- Email: [email protected]
Anisotropic fluctuations and stability of ordered phases in diblock copolymer systems
PHYSICA/,",,,, ELSEVIER
Physica A 244 (1997) 81 90
Anisotropic fluctuations and stability of ordered phases in diblock copolymer systems R a s h m i C. D e s a i * Department q[" Physics, University o]' Toronto, Toronto, Ont., Canada M5S IA7
Abstract
A review of our recent work on the stability of ordered phases in diblock copolymer systems is given. The stability is investigated and kinetic pathways of order-order transitions are explored using our novel theory of anisotropic fluctuations. The analogy between a polymer chain in periodic structures and electron in crystalline solids is discovered and exploited to obtain reciprocal space results. Anisotropic density-density correlation functions are computed for all ordered phases and used to obtain the spinodal lines in the weak segregation region of the lamellar, cylindrical and spherical phases. The most unstable fluctuation modes are identified. These in turn help in determining the kinetic pathways during the order-order transitions between the various phases. Dedicated to Professor Ben Widom on the occasion ~[ his 70th birthday
1. Introduction
Diblock copolymers consist of two (A and B) incompatible polymer chains which are chemically bonded. Because of their amphiphilic nature, they form a variety of ordered microphases at low tempeartures. The characteristic length scale of these microphases is dictated by the chain length, and leads to A- and B-rich domains separated by internal interfaces. The competition between the spontaneous curvature of the internal interfaces and the entropic stretching of the polymer chains determines the symmetry of the equilibrium microphases. Diblock copolymer melts are found to self-assemble into a variety of ordered microphases [1,2]. Diblock copolymers are characterized by the degree of polymerization, Z, and the volume fraction of the A-monomers, f . The interaction between the monomers is characterized by the Flory-Huggins parameter, Z. Traditionally, the phase diagram of a diblock copolymer melt is specified by the parameters z Z and f . The segregation between the two blocks is dictated by zZ, whereas * E-mail: [email protected]. 0378-4371/97/$17.00 Copyright @ 1997 Elsevier Science B.V. All rights reserved PII S 0 3 7 8 - 4 3 7 1 ( 9 7 ) 0 0 2 3 1 - 8
82
R.C. Desai/Physica A 244 (1997) 81-90
the interracial curvature is controlled by f . The order-disorder transition from the fully disordered homogeneous melt to the locally unmixed state occurs roughly at zZ = 10 for f = 31 [3]. As f is varied, for larger zZ, a variety of ordered phases including the classical lamellar(L), cylindrical(C) and spherical(S) structures are observed. Recently, three additional complex structures corresponding to hexagonally modulated or perforated layers (HML/HPL) [4], and a double gyroid (G) phase [5] have also been seen. Even though some understanding of the classical equilibrium phases have been obtained through mean field calculations [6], the thermodynamic stability of the ordered phases and the dynamical process through which they emerge remain relatively unexplored theoretically [7]. The lamellar to cylindrical transition is experimentally seen to proceed via an intermediate modulated layered state [8], and the cylinders were seen to transform into a lamellar structure directly in the experiments on triblock copolymers [9,10]. Recently, Hajduk et al. have shown that the HML and HPL morphologies are long-lived nonequilibrium states when the melt is driven away from the lamellar phase. In this mini-review, I summarise our recent work [ 11-13] on a theory for anisotropic fluctuations in ordered phases of diblock copolymer melts by means of a self-consistent expansion around the exact mean field solution, to quadratic (Gaussian) level. The self-consistent mean field theory neglects the composition fluctuations. Fredrickson and Helfand [14] extended Brasovskii theory [15] of fluctuations to diblock copolymers. It uses the Landau free energy functional of Leibler [3], and is valid in the weak segregation regime. Other similar theories which use isotropic correlation functions and Leibler's form of the free energy function include the work by Mayes and Olvera de la Cruz [16] and by Muthukumar [17]. By comparison, in our work the fluctuations are characterized by the anisotropic density~lensity correlation functions. We contend that in order to study the effect of fluctuations in an ordered brokensymmetry phase a self-consistent expansion must be performed arotmd the exact mean field solution for that ordered phase. We have developed such a theory [12,13], applied it to lamellar phase [11] identifying the most unstable mode, and now to all ordered phases [13]. The theory bears a useful and important analogy to the energy band theory in solids [12]. In crystalline solids, the basic problem is that of an electron in a periodic potential. Similarly, the chain conformations in diblock copolymeric systems are described by those of a flexible chain in a periodic potential arising from an ordered broken-symmetry phase. Both the mathematical problems correspond to obtaining eigenvalues and eigenfunctions of a diffusion equation in a periodic potential (the Schrrdinger equation is a diffusion equation with imaginary time). The analogy enables us to exploit the symmetry of the ordered phases, and to use many techniques developed in solid state physics. Using the anisotropic fluctuation theory, we have computed the scattering functions of all ordered phases, the spinodal lines, and the least stable or most unstable modes for each of the ordered phase during the order-order transitions in diblock copolymer
83
R.C. Desai/Physica A 244 (1997) 81 90
melts. We have also determined the bending and elastic moduli from the scattering functions for the lamellar phase. These results are also useful in infering the kinetic pathways for the order-order transitions and the epitaxial relations between various phases. In the next section, I give a brief overview of the theory. In Section 3, l give illustrative results and a brief discussion. For details, reader should consult the original references [1 1-13].
2. Theory of anisotropic fluctuations in diblock copolymer melts Consider a melt of JV" flexible AB-diblock copolymers in a volume V, with Z as polymerization number, f as volume fraction of A monomers, b as Kuhn length, v as segment volume, and Z as the Flory interaction parameter. All lengths are rescaled by radius of gyration, Rq = bv/Z/6. The entropic contribution to the free energy functional is described by propagator Q~(r, tlr' ), with ~ = A (B). It is the probability distribution of monomer t at r, given that monomer 0 is at r', and satisfies the equation [19] 8 Q~(r, tlrl ) = oCg~Q~(r,tlr, ) = (_~72 + ¢o~(r))Q~(r, tlr, )
5
with Q:(r, 0it' ) = 6(r - r'). Composition fluctuations are accounted for by expanding the volume fractions qS~= qS~) + 6~b~ and the auxiliary fields [18] o~ = o~°) + 6co~, around the exact mean field solutions in the ordered phases. Free energy of system can therefore be written as , ~ = ~,~(0) Jr- ~ ( I ) ~_ ~ ( 2 ) _[_ ....
The zeroth-order term is the mean field extremum, making first-order term vanish. The second-order term is obtained as
1 2
E~/s f dr dr'
C~¢(r,r')ao)~(r)6,o~(r')} ,
where C~ are integrals over four propagators:
CaA(r,r') =
1 fs' dt i dt' fdr, dr2dr3 Q}O) 0
0
× [o(~°~(r,,/A - ,it )o~°~(.,, - "lr')O~°'("."l*2)O(.°'(r2, + Q?~(.,./A
SBIr3 )
- tl*')O?~(r ', ' - t'i*)Q?~(., ,' 1.2)0'.°~(.2, YB i'3)],
R.C. Desai/Physica A 244 (1997) 81-90
84 ft
fl~
C A B ( r , r ' ) - Q~O) 1 /dtfdt'fdr, 0
drzdr3
0
× Q(A°)(r,, fA -- tlr)Q(A°)(r, tlr2 )p(B°)(r2,t'f)Q(2)(r ', f 8 - t'lr3 ). Thus, at the level of Gaussian fluctuations, the partition function becomes
After functional integration over 6co~(r), we obtain the effective free energy functional at a quadratic order in composition fluctuations:
F(2)({6~})= S
dr dr'[CeeA] -1 (r, r')6~b(r)6~b(r'),
where 6 ~ b - - 6 ~ A - 6~n and the two point density-density correlation function (in matrix form) is
c RPA-p°R3 I -
d
.~
2Z
'
where C = C - Aj • 12-1 • A2, and C = CAA -- CA~ -- CBA + CeB, Aj = CAA + CA. -- CBA -- CBB , A2 = CAA -- CAB + CBA -- CBB , Y. = CAA + CAB + CBA + C88.
Integrating out the composition fluctuations leads to the partition function
The mean field solution and the two-point cumulants are functions of the two parameters zZ and f only. The inclusion of fluctuations introduces a third parameter, 2 = poRg/Z 2 6 2 = (pZb6/63)Z, which quantifies fluctuation effects. Since the diblock copolymer problem requires the solution of diffusion equation which is nothing but a Schr6dinger equation in imaginary time, and the diblock copolymer melt is in an ordered state (the mean field o ~ ) is periodic), the problem is similar to an electronic system in a crystalline solid. Thus, we make use of Bloch's theorem.
R.C. DesailPhysica A 244 (1997) 81 90
85
The eigenmodes can be labeled by band index n and a wavevector k within the first brillouin Zone. Eigenfunctions have the form 1
~,
:~ / ( 2 .
i(k+G).r
G
where U']k(r)
Z U~k(G)exp(iG "r) G
and where the set {G} constitutes the reciprocal lattice vectors of the space group of the specified ordered phase. The problem requires the calculation of the coefficients {U,~k} and their eigenvalues {~:~k}, solution of the operator .)~(01 = __V2 _}_ ~o~0)(r).
Our study of the stability of an ordered phase proceeds as follows: (1) The mean field solution at a certain point of the phase diagram, ()~Z,f), is obtained by solving the self-consistent equations, in reciprocal space. The mean field calculations are used to obtain the phase diagram at weak segregations and to provide the Fourier components eJ~(G) needed for the fluctuations calculation. (2) The mean field Fourier components of m~(G) are then used to construct the Hamiltonian . ~ . The eigenvalues gnk and the components of their corresponding eigenvectors U,~k(G) are then calculated by solving the set of linear equations, at the wave vectors, k, which are constrained within the first Brillouin zone. We exploit the symmetries of the first Brillouin zone by considering only a portion of the Brillouin zone which is then discretized into a fine mesh. (3) The eigenvalues, e,, '~k, and eigenvectors, U,k, are then used to construct the matrix elements of C,,,,,,(k). (4) The eigenvalues of the I - (/Z/2)(~ are calculated. (5) The stability of the structure is investigated by examining the eigenvalues of ! - (zZ/2)C (the eigenvalues of Ig are always positive). If all the eigenvalues are positive, the phase is linearly stable or metastable. However, if at least one eigenvalue is negative, the phase is linearly unstable. The boundary between the metastable and the unstable regions defines the spinodal line.
3. Results and discusssion
The simplest application of the theory which reveals interesting physics of the problem is described in [12] where the case of weakly segregated symmetric diblock melt is considered, and the isotropic-lamellar transition studied. The problem reduces to the solution of the wellknown Mathieu differential equation. The parabolic eigenvalue spectrum for the homogeneous phase develops a band gap: for lamellar period L, and small
R.C. Desai/Physica A 244 (1997) 81-90
86
11.2
: :r
\,',, ,, 11.0 N
',3,', \'.'
/I/
'~ ,
,' i t /
\ \ ',',
,';H
'
''
~',
10.8
,'[
\, \',
,'/ ,4 '
/14
/ 1'7 ~ /,,I
/.7/
//
si
/) I'/
,,l ; / ,/../ //;,
10.6 10.4 0.40
0.45
0.50 f
0.55
0.60
Fig. 1. Phase diagram of diblock copolymer melts and the spinodal lines. See text for explanation of the figure.
composition fluctuation amplitude ¢, discontinuity occurs (in the first Brillouin zone) at kz = in/L, in the amount ;~¢. For a more general case [13], the broken symmetry of an ordered phase leads to formation of band yaps not present in the homogeneous phase. When one considers variation of the parameter pair (zZ, f ) , and moves from lamellar phase region towards the cylindrical phase, the eigenvalues 2nk of the operator C RPA often at the zone edge decrease to zero and become negative indicating onset of instability. Unstable mode(s) can be identified [11,13] for each order-order transition and the spinodal curves obtained in this manner. Fig. 1 displays the phase diagram in the (zZ, f ) space for diblock copolymer systems. Both the mean field phase boundaries and various spinodal lines are indicated. The three of the dark lines which meet at the consolute point are the mean field boundaries of the classical ordered phases: The dark solid line is the outer boundary for the lamellar phase, the cylindrical phase is contained in the region between the dark solid line and the dark short dashed line, and the spherical phase is in the region between the dark short dashed line and the dark long dashed line. Within the dark solid line, there are two spinodal lines: the thin dashed line (which also meets the consolute point) is the inner spinodal curve for cylindrical phase, and the dark dot-dashed line is the spinodal curve for the HPL phase. The latter does not reach the consolute point at its lowest zZ value. Within the cylindrical phase, there are also two spinodal lines: the thin solid line is the spinodal curve for the lamellar phase, and the thin long dashed line is the inner spinodal curve for the spherical phase. (The outer spinodal curve for the spherical phase lies in the disordered region and is not shown.) Finally, the thin short dashed line within the spherical phase region is the outer spinodal curve for the cylindrical phase. Thus, the one phase regions of the classical phases (L,C and S) are encapsulated by their spinodals: these are robust phases. The HPL phase is metastable in the centre of lamellar one-phase region and is unstable elsewhere. Experimentally observed HPL phase is in the region
R.C. Desai/Physica A 244 (1997) 81-90
87
of lamella~cylindrical-phase boundary, where our Gaussian fluctuation theory finds it thermodynamically unstable. Lamellar and HPL phases have equivalent topologies, and as a consequence symmetrical spinodal curves. The cylindrical and spherical phases on the other hand have different topology with majority and minority domains, leading to two symmetrically unrelated spinodal lines. From the fluctuation theory, we have constructed the scattering functions which describe the structure in Fourier space. The details are given in [11 13]. From the anisotropic scattering functions, one can also obtain information about the elastic properties of the copolymer blend [11]. The theory can also be used to construct a real space representation of the least stable fluctuation mode. Such figures show the interfaces between the A-rich and B-rich microdomains for the density profiles corresponding to different amplitudes of the unstable mode superposed over the mean field solution of an ordered phase at its order-order transition boundary. Figs. 2 and 3 show such contours, respectively, for the cylindrical to spherical and the cylindrical to lamellar transitions. In Fig. 2, for intermediate values of the unstable mode amplitude, fluctuations lead to bulges and necks in the cylinders. For sufficiently large amplitude of the unstable mode, the cylinders break up into spheres, the centres of which form the body-centered-cubic lattice with a periodicity identical to that of the equilibrium spherical phase, showing the epitaxial relation between the two ordered phases: cylinders are oriented along the [111] direction of the bcc lattice. In Fig. 3, the fluctuation modes lead to a flattening of the cylinders along the [01] direction for small fluctuation amplitudes, and lead for large amplitudes to the merging of the ellipsoidal cylinders to form modulated lamellae with a periodicity that is the same as the equilibrium lamellar phase at this point of' the phase diagram. In both cases, there are no intermediate states in agreement with the numerical study of Qi and Wang [20] and the experiments of Sakurai et al. [9,10]. A similar real space contours for lamellar to cylindrical transitions show that there can be intermediate states which can be metastable. This arises on account of the infinite degeneracy of the in-plane fluctuation modes which make the lamellar phase unstable. If one has a single unstable mode excited, there is a direct transition to cylindrical phase. In general however with two or more modes excited simultaneously leads to intermediate states that are modulated layered structures. Due to their convoluted nature. transformation to cylindrical structure requires a large transport of material which can create a rather slow kinetics. If the polymer melt is trapped in such an intermediate state, it may be identified as an 'equilibrium' phase in an experiment. In our analysis of the spherical to cylindrical transition, we also find intermediate modulated layered structures [13]. Thus, one of the conclusions of our theory of anisotropic fluctuations in diblock copolymer systems is that cylindrical-lamellar and cylindrical-spherical transitions are n o t reversible. It will be interesting to find experimental confirmation of these results. In summary, the inclusion of gaussian fluctuations around the e x a c t ordered mean field states in the theory for block copolymer systems leads to a wealth of new information and insights about the stability and kinetic pathways for order-order transitions in these systems.
R.C. Desai/Physica A 244 (1997) 81 90
88
Ca)
t .y
(b)
1
(c)
Fig. 2. Three-dimensional contour plots of the hexagonal phase, at zZ = 10.9, f = 0.43, close to the cylindrical-spherical phase boundary, showing the effects of the least stable mode on the ordered cylinders. The contour plots are defined by ~bA(r)= ~b0o)(r) + a ~-~, OSO)(k)~p,,~(t)= 0.5, where a is the amplitude of the least stable fluctuation mode. (a) corresponds to a = 0, i.e. to the mean field solution. (b) corresponds to a = 0.02. (c) corresponds to a = 0.075. Notice that the effect of a relatively large amplitude leads to the formation of spheres which are arranged in a body-centered cubic lattice. The [111] direction of the cubic unit cell of this lattice is along the cylindrical axis. Thus, the cylindrical to spherical phase transition is direct.
R.C. DesaiIPhysica A 244 (1997) 81 90
89
(a)
(b)
(c)
Fig. 3. Three-dimensional contour plots of the hexagonal phase, at z Z - 10.9, . / - 0 . 4 7 6 showing the cfl'ects of the least stable mode on the ordered cylinders: (a) corresponds to a - 0 , i.e. to the mean field solution; (b) corresponds to a = 0.06; (c) corresponds to a - 0.08. Notice that the effect of a relatively large amplitude leads to the formation of modulated lamellae parallel to the cylinders axes. Thus, the cylindrical to lamellar transition is direct without any intermediate state.
90
R. C Desai/Physica A 244 (1997) 81-90
Acknowledgements I have presented a mini-review of our recent work done with Professor Chuck Yeung, Dr. An-Chang Shi, Dr. Jaan Noolandi and Dr. Mohamed Laradji. It is a great pleasure to acknowledge their participation and collaboration. This work was also partially supported by Natural Sciences and Engineering Research Council of Canada.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20]
F.S. Bates, G.H. Fredrickson, Ann. Rev. Phys. Chem. 41 (1990) 525. F.S. Bates, M.F. Schulz, A.K. Khandpur, S.F. F6rster, J.H. Rosedale, Faraday Discuss. 98 (1994) 7. L. Leibler, Macromolecules 13 (1980) 1602. I.W. Hamley, K.A. Koppi, J.H. Rosedale, F.S. Bates, K. Almdal, K. Mortensen, Macromolecules 26 (1993) 5959. D.A. Hajduk, P.E. Harper, S.M. Gruner, C.C. Honeker~ G. Kim, E.L. Thomas, L.J. Fetters, Macromolecules 27 (1994) 4063. M.W. Matsen, F.S. Bates, Macromolecules 29 (1996) 1091. R.E. Cohen, F.S. Bates, J. Polym. Sei.: Polym. Phys. 18 (1980) 2143. D.A. Hajduk, SM. Gruner, P. Rangarajan, R.A. Register, L.J. Fetters, C.C. Honeker, R.J. Albalak, E.L. Thomas, Macromolecules 27 (1994) 490. S. Sakurai, T. Momii, K. Taie, M. Shibayama, S. Nomura, T. Hashimoto, Macromolecules 26 (1993) 485. S. Sakurai, H. Umeda, K. Yaie, S. Nomura, J. Chem. Phys. 105 (1996) 8902. C. Yeung, A.-C. Shi, J. Noolandi, R.C. Desai, Macromol. Theory Simul. 5 (1996) 291. A.-C. Shi, J. Noolandi, R.C. Desai, Macromolecules 29 (1996) 6487. M. Laradji, A.-C. Shi, J. Noolandi, R.C. Desai, Macromolecules, in press; Phys. Rev. Lett. 78 (1997) 2577. G.H. Fredrickson, E. Helfand, J. Chem. Phys. 87 (1987) 697. S.A. Brasovskii, Zh. Eksp. Teor. Fiz. 68 (1975) 175 [Soy. Phys. JETP 41 (1975) 85]. A.M. Mayes, M. Olvera de la Cruz, J. Chem. Phys. 95 (1991) 1670. M. Muthukumar, Macromolecules 26 (1993) 5259. K.M. Hong, J. Noolandi, Macromolecules 14 (1981) 727. M. Doi, S.F. Edwards, The Theory of Polymer Dynamics, Oxford University Press, London, 1986. S. Qi, Z.-G. Wang, Phys. Rev. Lett. 76 (1996) 1679; Phys. Rev. E, in press.
Physica A 244 (1997) 81 90
Anisotropic fluctuations and stability of ordered phases in diblock copolymer systems R a s h m i C. D e s a i * Department q[" Physics, University o]' Toronto, Toronto, Ont., Canada M5S IA7
Abstract
A review of our recent work on the stability of ordered phases in diblock copolymer systems is given. The stability is investigated and kinetic pathways of order-order transitions are explored using our novel theory of anisotropic fluctuations. The analogy between a polymer chain in periodic structures and electron in crystalline solids is discovered and exploited to obtain reciprocal space results. Anisotropic density-density correlation functions are computed for all ordered phases and used to obtain the spinodal lines in the weak segregation region of the lamellar, cylindrical and spherical phases. The most unstable fluctuation modes are identified. These in turn help in determining the kinetic pathways during the order-order transitions between the various phases. Dedicated to Professor Ben Widom on the occasion ~[ his 70th birthday
1. Introduction
Diblock copolymers consist of two (A and B) incompatible polymer chains which are chemically bonded. Because of their amphiphilic nature, they form a variety of ordered microphases at low tempeartures. The characteristic length scale of these microphases is dictated by the chain length, and leads to A- and B-rich domains separated by internal interfaces. The competition between the spontaneous curvature of the internal interfaces and the entropic stretching of the polymer chains determines the symmetry of the equilibrium microphases. Diblock copolymer melts are found to self-assemble into a variety of ordered microphases [1,2]. Diblock copolymers are characterized by the degree of polymerization, Z, and the volume fraction of the A-monomers, f . The interaction between the monomers is characterized by the Flory-Huggins parameter, Z. Traditionally, the phase diagram of a diblock copolymer melt is specified by the parameters z Z and f . The segregation between the two blocks is dictated by zZ, whereas * E-mail: [email protected]. 0378-4371/97/$17.00 Copyright @ 1997 Elsevier Science B.V. All rights reserved PII S 0 3 7 8 - 4 3 7 1 ( 9 7 ) 0 0 2 3 1 - 8
82
R.C. Desai/Physica A 244 (1997) 81-90
the interracial curvature is controlled by f . The order-disorder transition from the fully disordered homogeneous melt to the locally unmixed state occurs roughly at zZ = 10 for f = 31 [3]. As f is varied, for larger zZ, a variety of ordered phases including the classical lamellar(L), cylindrical(C) and spherical(S) structures are observed. Recently, three additional complex structures corresponding to hexagonally modulated or perforated layers (HML/HPL) [4], and a double gyroid (G) phase [5] have also been seen. Even though some understanding of the classical equilibrium phases have been obtained through mean field calculations [6], the thermodynamic stability of the ordered phases and the dynamical process through which they emerge remain relatively unexplored theoretically [7]. The lamellar to cylindrical transition is experimentally seen to proceed via an intermediate modulated layered state [8], and the cylinders were seen to transform into a lamellar structure directly in the experiments on triblock copolymers [9,10]. Recently, Hajduk et al. have shown that the HML and HPL morphologies are long-lived nonequilibrium states when the melt is driven away from the lamellar phase. In this mini-review, I summarise our recent work [ 11-13] on a theory for anisotropic fluctuations in ordered phases of diblock copolymer melts by means of a self-consistent expansion around the exact mean field solution, to quadratic (Gaussian) level. The self-consistent mean field theory neglects the composition fluctuations. Fredrickson and Helfand [14] extended Brasovskii theory [15] of fluctuations to diblock copolymers. It uses the Landau free energy functional of Leibler [3], and is valid in the weak segregation regime. Other similar theories which use isotropic correlation functions and Leibler's form of the free energy function include the work by Mayes and Olvera de la Cruz [16] and by Muthukumar [17]. By comparison, in our work the fluctuations are characterized by the anisotropic density~lensity correlation functions. We contend that in order to study the effect of fluctuations in an ordered brokensymmetry phase a self-consistent expansion must be performed arotmd the exact mean field solution for that ordered phase. We have developed such a theory [12,13], applied it to lamellar phase [11] identifying the most unstable mode, and now to all ordered phases [13]. The theory bears a useful and important analogy to the energy band theory in solids [12]. In crystalline solids, the basic problem is that of an electron in a periodic potential. Similarly, the chain conformations in diblock copolymeric systems are described by those of a flexible chain in a periodic potential arising from an ordered broken-symmetry phase. Both the mathematical problems correspond to obtaining eigenvalues and eigenfunctions of a diffusion equation in a periodic potential (the Schrrdinger equation is a diffusion equation with imaginary time). The analogy enables us to exploit the symmetry of the ordered phases, and to use many techniques developed in solid state physics. Using the anisotropic fluctuation theory, we have computed the scattering functions of all ordered phases, the spinodal lines, and the least stable or most unstable modes for each of the ordered phase during the order-order transitions in diblock copolymer
83
R.C. Desai/Physica A 244 (1997) 81 90
melts. We have also determined the bending and elastic moduli from the scattering functions for the lamellar phase. These results are also useful in infering the kinetic pathways for the order-order transitions and the epitaxial relations between various phases. In the next section, I give a brief overview of the theory. In Section 3, l give illustrative results and a brief discussion. For details, reader should consult the original references [1 1-13].
2. Theory of anisotropic fluctuations in diblock copolymer melts Consider a melt of JV" flexible AB-diblock copolymers in a volume V, with Z as polymerization number, f as volume fraction of A monomers, b as Kuhn length, v as segment volume, and Z as the Flory interaction parameter. All lengths are rescaled by radius of gyration, Rq = bv/Z/6. The entropic contribution to the free energy functional is described by propagator Q~(r, tlr' ), with ~ = A (B). It is the probability distribution of monomer t at r, given that monomer 0 is at r', and satisfies the equation [19] 8 Q~(r, tlrl ) = oCg~Q~(r,tlr, ) = (_~72 + ¢o~(r))Q~(r, tlr, )
5
with Q:(r, 0it' ) = 6(r - r'). Composition fluctuations are accounted for by expanding the volume fractions qS~= qS~) + 6~b~ and the auxiliary fields [18] o~ = o~°) + 6co~, around the exact mean field solutions in the ordered phases. Free energy of system can therefore be written as , ~ = ~,~(0) Jr- ~ ( I ) ~_ ~ ( 2 ) _[_ ....
The zeroth-order term is the mean field extremum, making first-order term vanish. The second-order term is obtained as
1 2
E~/s f dr dr'
C~¢(r,r')ao)~(r)6,o~(r')} ,
where C~ are integrals over four propagators:
CaA(r,r') =
1 fs' dt i dt' fdr, dr2dr3 Q}O) 0
0
× [o(~°~(r,,/A - ,it )o~°~(.,, - "lr')O~°'("."l*2)O(.°'(r2, + Q?~(.,./A
SBIr3 )
- tl*')O?~(r ', ' - t'i*)Q?~(., ,' 1.2)0'.°~(.2, YB i'3)],
R.C. Desai/Physica A 244 (1997) 81-90
84 ft
fl~
C A B ( r , r ' ) - Q~O) 1 /dtfdt'fdr, 0
drzdr3
0
× Q(A°)(r,, fA -- tlr)Q(A°)(r, tlr2 )p(B°)(r2,t'f)Q(2)(r ', f 8 - t'lr3 ). Thus, at the level of Gaussian fluctuations, the partition function becomes
After functional integration over 6co~(r), we obtain the effective free energy functional at a quadratic order in composition fluctuations:
F(2)({6~})= S
dr dr'[CeeA] -1 (r, r')6~b(r)6~b(r'),
where 6 ~ b - - 6 ~ A - 6~n and the two point density-density correlation function (in matrix form) is
c RPA-p°R3 I -
d
.~
2Z
'
where C = C - Aj • 12-1 • A2, and C = CAA -- CA~ -- CBA + CeB, Aj = CAA + CA. -- CBA -- CBB , A2 = CAA -- CAB + CBA -- CBB , Y. = CAA + CAB + CBA + C88.
Integrating out the composition fluctuations leads to the partition function
The mean field solution and the two-point cumulants are functions of the two parameters zZ and f only. The inclusion of fluctuations introduces a third parameter, 2 = poRg/Z 2 6 2 = (pZb6/63)Z, which quantifies fluctuation effects. Since the diblock copolymer problem requires the solution of diffusion equation which is nothing but a Schr6dinger equation in imaginary time, and the diblock copolymer melt is in an ordered state (the mean field o ~ ) is periodic), the problem is similar to an electronic system in a crystalline solid. Thus, we make use of Bloch's theorem.
R.C. DesailPhysica A 244 (1997) 81 90
85
The eigenmodes can be labeled by band index n and a wavevector k within the first brillouin Zone. Eigenfunctions have the form 1
~,
:~ / ( 2 .
i(k+G).r
G
where U']k(r)
Z U~k(G)exp(iG "r) G
and where the set {G} constitutes the reciprocal lattice vectors of the space group of the specified ordered phase. The problem requires the calculation of the coefficients {U,~k} and their eigenvalues {~:~k}, solution of the operator .)~(01 = __V2 _}_ ~o~0)(r).
Our study of the stability of an ordered phase proceeds as follows: (1) The mean field solution at a certain point of the phase diagram, ()~Z,f), is obtained by solving the self-consistent equations, in reciprocal space. The mean field calculations are used to obtain the phase diagram at weak segregations and to provide the Fourier components eJ~(G) needed for the fluctuations calculation. (2) The mean field Fourier components of m~(G) are then used to construct the Hamiltonian . ~ . The eigenvalues gnk and the components of their corresponding eigenvectors U,~k(G) are then calculated by solving the set of linear equations, at the wave vectors, k, which are constrained within the first Brillouin zone. We exploit the symmetries of the first Brillouin zone by considering only a portion of the Brillouin zone which is then discretized into a fine mesh. (3) The eigenvalues, e,, '~k, and eigenvectors, U,k, are then used to construct the matrix elements of C,,,,,,(k). (4) The eigenvalues of the I - (/Z/2)(~ are calculated. (5) The stability of the structure is investigated by examining the eigenvalues of ! - (zZ/2)C (the eigenvalues of Ig are always positive). If all the eigenvalues are positive, the phase is linearly stable or metastable. However, if at least one eigenvalue is negative, the phase is linearly unstable. The boundary between the metastable and the unstable regions defines the spinodal line.
3. Results and discusssion
The simplest application of the theory which reveals interesting physics of the problem is described in [12] where the case of weakly segregated symmetric diblock melt is considered, and the isotropic-lamellar transition studied. The problem reduces to the solution of the wellknown Mathieu differential equation. The parabolic eigenvalue spectrum for the homogeneous phase develops a band gap: for lamellar period L, and small
R.C. Desai/Physica A 244 (1997) 81-90
86
11.2
: :r
\,',, ,, 11.0 N
',3,', \'.'
/I/
'~ ,
,' i t /
\ \ ',',
,';H
'
''
~',
10.8
,'[
\, \',
,'/ ,4 '
/14
/ 1'7 ~ /,,I
/.7/
//
si
/) I'/
,,l ; / ,/../ //;,
10.6 10.4 0.40
0.45
0.50 f
0.55
0.60
Fig. 1. Phase diagram of diblock copolymer melts and the spinodal lines. See text for explanation of the figure.
composition fluctuation amplitude ¢, discontinuity occurs (in the first Brillouin zone) at kz = in/L, in the amount ;~¢. For a more general case [13], the broken symmetry of an ordered phase leads to formation of band yaps not present in the homogeneous phase. When one considers variation of the parameter pair (zZ, f ) , and moves from lamellar phase region towards the cylindrical phase, the eigenvalues 2nk of the operator C RPA often at the zone edge decrease to zero and become negative indicating onset of instability. Unstable mode(s) can be identified [11,13] for each order-order transition and the spinodal curves obtained in this manner. Fig. 1 displays the phase diagram in the (zZ, f ) space for diblock copolymer systems. Both the mean field phase boundaries and various spinodal lines are indicated. The three of the dark lines which meet at the consolute point are the mean field boundaries of the classical ordered phases: The dark solid line is the outer boundary for the lamellar phase, the cylindrical phase is contained in the region between the dark solid line and the dark short dashed line, and the spherical phase is in the region between the dark short dashed line and the dark long dashed line. Within the dark solid line, there are two spinodal lines: the thin dashed line (which also meets the consolute point) is the inner spinodal curve for cylindrical phase, and the dark dot-dashed line is the spinodal curve for the HPL phase. The latter does not reach the consolute point at its lowest zZ value. Within the cylindrical phase, there are also two spinodal lines: the thin solid line is the spinodal curve for the lamellar phase, and the thin long dashed line is the inner spinodal curve for the spherical phase. (The outer spinodal curve for the spherical phase lies in the disordered region and is not shown.) Finally, the thin short dashed line within the spherical phase region is the outer spinodal curve for the cylindrical phase. Thus, the one phase regions of the classical phases (L,C and S) are encapsulated by their spinodals: these are robust phases. The HPL phase is metastable in the centre of lamellar one-phase region and is unstable elsewhere. Experimentally observed HPL phase is in the region
R.C. Desai/Physica A 244 (1997) 81-90
87
of lamella~cylindrical-phase boundary, where our Gaussian fluctuation theory finds it thermodynamically unstable. Lamellar and HPL phases have equivalent topologies, and as a consequence symmetrical spinodal curves. The cylindrical and spherical phases on the other hand have different topology with majority and minority domains, leading to two symmetrically unrelated spinodal lines. From the fluctuation theory, we have constructed the scattering functions which describe the structure in Fourier space. The details are given in [11 13]. From the anisotropic scattering functions, one can also obtain information about the elastic properties of the copolymer blend [11]. The theory can also be used to construct a real space representation of the least stable fluctuation mode. Such figures show the interfaces between the A-rich and B-rich microdomains for the density profiles corresponding to different amplitudes of the unstable mode superposed over the mean field solution of an ordered phase at its order-order transition boundary. Figs. 2 and 3 show such contours, respectively, for the cylindrical to spherical and the cylindrical to lamellar transitions. In Fig. 2, for intermediate values of the unstable mode amplitude, fluctuations lead to bulges and necks in the cylinders. For sufficiently large amplitude of the unstable mode, the cylinders break up into spheres, the centres of which form the body-centered-cubic lattice with a periodicity identical to that of the equilibrium spherical phase, showing the epitaxial relation between the two ordered phases: cylinders are oriented along the [111] direction of the bcc lattice. In Fig. 3, the fluctuation modes lead to a flattening of the cylinders along the [01] direction for small fluctuation amplitudes, and lead for large amplitudes to the merging of the ellipsoidal cylinders to form modulated lamellae with a periodicity that is the same as the equilibrium lamellar phase at this point of' the phase diagram. In both cases, there are no intermediate states in agreement with the numerical study of Qi and Wang [20] and the experiments of Sakurai et al. [9,10]. A similar real space contours for lamellar to cylindrical transitions show that there can be intermediate states which can be metastable. This arises on account of the infinite degeneracy of the in-plane fluctuation modes which make the lamellar phase unstable. If one has a single unstable mode excited, there is a direct transition to cylindrical phase. In general however with two or more modes excited simultaneously leads to intermediate states that are modulated layered structures. Due to their convoluted nature. transformation to cylindrical structure requires a large transport of material which can create a rather slow kinetics. If the polymer melt is trapped in such an intermediate state, it may be identified as an 'equilibrium' phase in an experiment. In our analysis of the spherical to cylindrical transition, we also find intermediate modulated layered structures [13]. Thus, one of the conclusions of our theory of anisotropic fluctuations in diblock copolymer systems is that cylindrical-lamellar and cylindrical-spherical transitions are n o t reversible. It will be interesting to find experimental confirmation of these results. In summary, the inclusion of gaussian fluctuations around the e x a c t ordered mean field states in the theory for block copolymer systems leads to a wealth of new information and insights about the stability and kinetic pathways for order-order transitions in these systems.
R.C. Desai/Physica A 244 (1997) 81 90
88
Ca)
t .y
(b)
1
(c)
Fig. 2. Three-dimensional contour plots of the hexagonal phase, at zZ = 10.9, f = 0.43, close to the cylindrical-spherical phase boundary, showing the effects of the least stable mode on the ordered cylinders. The contour plots are defined by ~bA(r)= ~b0o)(r) + a ~-~, OSO)(k)~p,,~(t)= 0.5, where a is the amplitude of the least stable fluctuation mode. (a) corresponds to a = 0, i.e. to the mean field solution. (b) corresponds to a = 0.02. (c) corresponds to a = 0.075. Notice that the effect of a relatively large amplitude leads to the formation of spheres which are arranged in a body-centered cubic lattice. The [111] direction of the cubic unit cell of this lattice is along the cylindrical axis. Thus, the cylindrical to spherical phase transition is direct.
R.C. DesaiIPhysica A 244 (1997) 81 90
89
(a)
(b)
(c)
Fig. 3. Three-dimensional contour plots of the hexagonal phase, at z Z - 10.9, . / - 0 . 4 7 6 showing the cfl'ects of the least stable mode on the ordered cylinders: (a) corresponds to a - 0 , i.e. to the mean field solution; (b) corresponds to a = 0.06; (c) corresponds to a - 0.08. Notice that the effect of a relatively large amplitude leads to the formation of modulated lamellae parallel to the cylinders axes. Thus, the cylindrical to lamellar transition is direct without any intermediate state.
90
R. C Desai/Physica A 244 (1997) 81-90
Acknowledgements I have presented a mini-review of our recent work done with Professor Chuck Yeung, Dr. An-Chang Shi, Dr. Jaan Noolandi and Dr. Mohamed Laradji. It is a great pleasure to acknowledge their participation and collaboration. This work was also partially supported by Natural Sciences and Engineering Research Council of Canada.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20]
F.S. Bates, G.H. Fredrickson, Ann. Rev. Phys. Chem. 41 (1990) 525. F.S. Bates, M.F. Schulz, A.K. Khandpur, S.F. F6rster, J.H. Rosedale, Faraday Discuss. 98 (1994) 7. L. Leibler, Macromolecules 13 (1980) 1602. I.W. Hamley, K.A. Koppi, J.H. Rosedale, F.S. Bates, K. Almdal, K. Mortensen, Macromolecules 26 (1993) 5959. D.A. Hajduk, P.E. Harper, S.M. Gruner, C.C. Honeker~ G. Kim, E.L. Thomas, L.J. Fetters, Macromolecules 27 (1994) 4063. M.W. Matsen, F.S. Bates, Macromolecules 29 (1996) 1091. R.E. Cohen, F.S. Bates, J. Polym. Sei.: Polym. Phys. 18 (1980) 2143. D.A. Hajduk, SM. Gruner, P. Rangarajan, R.A. Register, L.J. Fetters, C.C. Honeker, R.J. Albalak, E.L. Thomas, Macromolecules 27 (1994) 490. S. Sakurai, T. Momii, K. Taie, M. Shibayama, S. Nomura, T. Hashimoto, Macromolecules 26 (1993) 485. S. Sakurai, H. Umeda, K. Yaie, S. Nomura, J. Chem. Phys. 105 (1996) 8902. C. Yeung, A.-C. Shi, J. Noolandi, R.C. Desai, Macromol. Theory Simul. 5 (1996) 291. A.-C. Shi, J. Noolandi, R.C. Desai, Macromolecules 29 (1996) 6487. M. Laradji, A.-C. Shi, J. Noolandi, R.C. Desai, Macromolecules, in press; Phys. Rev. Lett. 78 (1997) 2577. G.H. Fredrickson, E. Helfand, J. Chem. Phys. 87 (1987) 697. S.A. Brasovskii, Zh. Eksp. Teor. Fiz. 68 (1975) 175 [Soy. Phys. JETP 41 (1975) 85]. A.M. Mayes, M. Olvera de la Cruz, J. Chem. Phys. 95 (1991) 1670. M. Muthukumar, Macromolecules 26 (1993) 5259. K.M. Hong, J. Noolandi, Macromolecules 14 (1981) 727. M. Doi, S.F. Edwards, The Theory of Polymer Dynamics, Oxford University Press, London, 1986. S. Qi, Z.-G. Wang, Phys. Rev. Lett. 76 (1996) 1679; Phys. Rev. E, in press.