3.2 Understanding DSA using a continuum model
4.1.2 Fluctuation modes of defect-free structures
5 6
0 500 1000 1500 2000 2500 3000 3500
λk or Σi vk,i / 2000
k λk
H
Σi vk,i / 2000 λk
G
-0.05 0 0.05
0 10 20
Figure 4.5: Average of every eigenvector (red) and the corresponding eigenval-ues (gray) of the transformed Hessian matrix G. The eigenvector average is rescaled asP
i
vki /2000 to facilitate visualization.
4.1.2 Fluctuation modes of defect-free structures
With a valid set of eigenvectors representing the local changes of the order pa-rameter, it is possible to describe the local fluctuations associated with every eigenvector of that space.
It has been previously shown that fluctuations associated to modes with λk = 0 do not change the free-energy of the corresponding stable state, which is the case for modesk=0 andk =1 (see figure4.6). Modek =0 is discarded from the analysis since the associated fluctuations violate mass conservation.
To understand the nature of mode k = 1, it is useful to recall that fluctua-tions where∆F=0 are related to the continuous symmetries of the free-energy functional. The Ohta-Kawasaki model is translational and rotational invariant;
however, the translation,F[m(x)] =F[m(x+ε)], is the only continuous symme-try. For a given system size, only certain rotations ofm(x)leave the free energy unchanged. That is the case for 90-degree rotations, and multiples of it, in a square 2D system commensurate to the equilibrium spacing LO, i.e., a system with LY = LX = n LO. Therefore rotation is not a continuous symmetry of the
0 2 4 LX/LO
0 L/LYO 2
k= 0 λk= 0.00000 0.40
0
-0.40 δm(x)
0 2 4
LX/LO
0 L/LYO 2
k= 1 λk= -0.00000
-0.68 0 0.68
δm(x)
Figure 4.6: First two eigenvectors (color map) of the transformed Hessian ma-trixGassociated with no change in the free energy during a perturbation. The modek = 0 corresponds to constant change of the local composition and fails to satisfy mass conservation. Therefore, changes in the configuration due to this mode provide no physical insights. Black lines signify the AB interfaces ofmo. free-energy functional.
Provided that a fluctuation mode corresponding to the translational symme-try exists, the following expression must be satisfied.
m(x+ε) =m(x) +εek(x) ek(x) = m(x+ε) −m(x)
ε
This expression corresponds to a simple numerical approximation for the first-order derivative, dm/dx. Figure4.7 confirms that modek = 1 corresponds to the first-order derivative of the lamella configuration mo by comparing their profiles along theX direction. Moreover, provided that translation is a contin-uous symmetry of this model, the eigenvalue of such mode must be zero. This is the case for mode k = 1 and therefore this eigenvector corresponds to the translation mode of the lamella configuration.
We now proceed to investigate other fluctuation modes that can be expected due to broken symmetries. The formation of ordered phases of block-copolymers from the disordered state resembles the crystallization process observed in hard crystals. This trait has been exploited to draw parallels and investigate these soft matter systems building on the extensive knowledge of atomic crystals [102].
An important difference, however, must be highlighted: in self-assembled phases of block copolymers, it is the collective density what undergoes a crystallization-type ordering and not the coordinates of individual particlesi. This is evident
iWe refer to particles recalling that the collective density description in a continuum model correspond to the underlying particle based description of a block copolymer chain.
0 1 LY/LO m(x) 0
0 e(x)k
Figure 4.7: Comparison of the composition profile alongXfor the lamella con-figuration (red line), shown in figure 4.1, with the eigenvector profile corre-sponding to the modek = 1 (blue line), shown in figure4.6. The profile of the eigenvector,ek(x), corresponds to the first derivative of the composition profile, m(x), thus confirming that this vector describes a translation mode.
in the choice of the order parameter: for the soft matter system of our inter-est, the order parameter is the local densities of the A or B blocks of the poly-mer chains, whereas in crystals it is the local displacements of atoms from their lowest-energy lattice positions. Nevertheless, a useful parallel can be drawn be-tween the fluctuation modes of our system (i.e., the eigenvectors of the Hessian matrix of the free energy of the block-copolymer system) and the phonons (i.e., the eigenvectors of the potential energy) representing the elementary energetic fluctuations in crystals.
Phonons and the broken symmetries in crystals are related via the Goldstone theorem; due to the spontaneous breaking of the continuous translational sym-metry, the system exhibits low-energy excitations which have low-frequencies, i.e., Goldstone modes [103, 104]. Thus, the low-frequency phonons in crystals correspond to the Goldstone modes for the translations; whose associated en-ergy cost vanishes in the limit of infinite wavelength [105, 106]
The free-energy cost associated to the deformation of the AB interfaces,∆F, has been previously described for the case of small-amplitude undulations as [107]
∆F=F−Fo' Z
dr γ
2 [∇δx(y)]2+K
2 [4δx(y)]2
+ Z
dr B
2 [δx(y)]2 (4.16) where Fo corresponds to the free energy of the lamella without deformation, δx(y)is the amplitude of the undulation along the interface, andKandBare the bending and compression moduli, respectively. The first term on the right-hand
side of the above equation accounts for the energy of the capillary waves where the lamellar structure undergoes a collective bending, i.e., the undulation waves are in-phase. In Fourier space, this first term has the form, ∆F ∼ γq2Y +Kq4Y, whereqY is the wave vector in the direction parallel to the interfaces. When the undulation waves are out of phase, the domains will be compressed or stretched and the corresponding free-energy cost will be described by the second term on the right-hand side of Eq.4.16.
In this regard, we investigate the variations of the lowest energy fluctuation modes due to changes in the wave vector,qY(k) = 2πk/LY. The smallest non-zero eigenvaluesλk in the spectrum correspond to the modes with the lowest frequencies, i.e., the longest wavelength; therefore, we investigate modes 2 6 k 6 7 (cf. figure 4.5). These modes are degenerate such that we express the changes ofλk as a function ofLY only for modesk=2, 4, 7.
Figure4.8confirms that as the wave vectorqY(k)tends to zero, i.e.,LY tends to infinity, the frequency of modesk=2, 4 approaches zero as well and their ex-cess free energy vanishes (cf. eq.4.6). In that limit, these two fluctuation modes approach the translation mode of the configuration, which has an excess free energy of zero as previously discussed. Thus, modesk=2, 4 are the Goldstone excitations of the lamella due to the spontaneous breaking of the translational symmetry. In contrast, the peristaltic fluctuation due to modek=6 requires the compression of the lamellar phase with an associated excess free energy∆F>0 regardless of the wave vectorqY(k).
Furthermore, by resorting to the capillary wave theory of interfaces [108], these Goldstone modes identified in the continuum description of the block-copolymer system of our work,k = 2, 4, correspond to the thermally excited capillary waves that strongly influence the interface width in block copolymer systems [105, 109, 110].
Fluctuations of the configurations corresponding to a given mode are better described by computing directly the perturbed configurationsm(x) =mo(x) + εek(x) where ε can take any real value. In figure 4.9, therefore, we consider fluctuations of the same magnitude but opposite sign to show snapshots of the perturbation of the lamella phase with four different modes.
We corroborate that modek=0 corresponds to the translation of the config-uration in the direction perpendicular to the lamella, which is natural in view of the symmetry of the system. It is clear now that no free-energy change is as-sociated with this fluctuation. Furthermore, the soft modek=2 corresponds to an undulation of the AB interfaces and the same type of fluctuation is observed with the corresponding degenerate mode k = 3, but the undulation wave has a different phase in this case. These three first normal modes correspond to fluctuations arising from the collective motion of the lamella.
(a)
0 LX/LO
0 LY/LO
k= 2 λk= 1.816e-03
L0X/LO 0 2 4 6
LY/LO
k= 2 λk= 2.067e-06
(b)
k= 6 k= 4
k= 2
(c)
0 0.02 0.04 0.06 0.08 0.1 0.12
0 0.2 0.4 0.6 0.8 1
λk
1 / (LY/LO)2 k = 2
k = 4 k = 6
Figure 4.8: Goldstone modes in lamellar phase mo. (a) The wavelength of the undulation increases as the wave vector, qY(k) = 2πk/LY, decays. (b) Low-energy fluctuation modes (color map) corresponding to the lowest non-zero eigenvalues. The deformation of the AB interfaces (red contour line) arising from the perturbation of the lamellar phase in equilibrium (black contour line) allows to differentiate the peristaltic motion from the lamellar undulations. (c) The frequency of the peristaltic motion remains constant regardless of wave vector qY(k). In contrast, the frequency vanishes asqY(k) → 0 for the low-line undulations, k = 2, 4. The later indicate that these modes correspond to the Goldstone excitations originated by the broken translational symmetry.
Figure 4.9: Fluctuations of the lamellar phase due to four different modes. (a) A fluctuation mode describes the changes in the local composition,δm(x), which in the case of a soft mode (color map) will cause fluctuations of the AB interfaces of the corresponding unperturbed lamellamo (black contour lines). (b) Mode k = 1 depicts changes associated with the translation of the configuration mo. Modesk = 2, 3 are degenerate soft modes associated to the lowest free-energy change. Modek=3000, at the last part of the spectrum of eigenvalues, depicts high-frequency local fluctuations. (c) Perturbations of the same magnitude but opposite sign, where white represents the AB interfaces. A value of |ε| = 1 is used for better visualization.
In contrast, as the eigenvalue increases and thus the spatial frequency of the undulations increases the fluctuations correspond to more localized changes.
Modek=3000 corresponds to a much higher fluctuation, close to the end of the spectrum, where undulations do not correspond to a displacement of internal AB interfaces but rather to local composition fluctuations inside the domains.
It is worth pointing out that in the course of the transition of the lamella configurationmotowards another given stable state, the key changes will occur on the AB interfaces. Therefore, we focus our attention on the behavior of such interfaces to investigate the various fluctuation modes.
The lowest fluctuation modes in our study, e.g.,k = 9, 13, 15, correspond to capillary waves in 2D as depicted in red contour lines in figure4.10. Despite of having rather close eigenvalues, i.e., a similar wavelength of the undulations, the coupling of the fluctuation of each AB interface in the lamellar phase is dif-ferent in every mode, thus indicating that they are not degenerate. The smallest difference between their values suggests that two given eigenvalues,λi andλj will be degenerate if ∆λ 6 4x10−5 for the collocation grid deployed in these calculations.
In this regard, the fluctuation modes offer a complementary description of the AB interfaces in the continuum model, provided that such modes agree with the capillary wave description of interfaces and satisfy the constraints imposed by the symmetry of the free energy functional: the existence of modes describ-ing Goldstone-type excitations and a mode related to the translation of the con-figuration.