10 8 6 4 3 2
1
1 10 100 1000
λ /N 0.5
χ N
N = 62 N = 120 N = 210 N = 300
Figure 3.13: Scaling plot of lamellar distance λ depending on χ and N for four chain lengths 62,120,210,300 in the strong segregation-regime. The dashed straight line has a slope n ≃ 0.22.
Data points for N = 120 are continued to lower χN-values to regain the weak–segragation behavior λ ∼ N0.5. All data were obtained in simulations with minimization of the free energy (2.2) with (2.15), (2.16).
their model.
3.5 Asymmetric Chains
We also carried out bulk simulations for asymmetric chains with the length N = 100 and the A-monomer fraction fA = 0.3. The existence of a cylindrical phase is exemplified by
0.01 0.1 1 10 100 1000 10000
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
S(k)
k
40 MCS 4 × 10
24×10
310
5Figure 3.14: The structure factor developing in time in the melt of symmetric chains with the length N = 120. The initial configuration was a disordered melt at the infinite temperatureχ= 0. At t= 0 the mismatch parameter χ was set to 0.45. Note that the structure factor has a peak even att = 0 corresponding to the mean radius of gyration of A blocks.
Fig. 3.16 where superstructure peaks of the spherically averaged structure factor of the minority component characteristic for this structure are clearly seen. To obtain Fig. 3.16, we used χN = 65.0. In Fig. 3.17 we plot the isosurfaces for this structure. One can clearly observe the columnar tubes of the minority component A ordered on a hexagonal 2−d lattice. Some connections between neighboring tubes seen in the figure are actually additional domains of cylindrical structure with their orientations different from that of the main domain. The whole multidomain structure remains stable when the simulation going further on which may be attributed to the absence of hydrodynamic flows in the Monte Carlo algorithm.
In Fig. 3.18 we present isosurfaces obtained in simulations on asymmetric chains fA = 0.17 (chain length N = 150) at χN = 45.0. The minority component is contained in spheres or bulbs ordered in a body–centered cubic (bcc) lattice. There are still some A-blocks in the B-rich region which do not disappear as the simulation goes on.
3.5 Asymmetric Chains
1.0 0.8 0.6 0.4
0.2
0.1
10
110
210
310
4k*
t
Figure 3.15: The position of the peak of the structure factor as estimated with the Lorenz fit versus time in the melt of symmetric chainsN = 120 after suddenly increasingχ from zero to 0.45. The line represents a power law fit with the exponent 0.15 to the data at the initial times.
10 -1 10 0 10 1 10 2 10 3 10 4
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
S(k)
k 1
3 1/2
7 1/2
f A = 0.3 χ N = 65 1
3 1/2
7 1/2
f A = 0.3 χ N = 65
Figure 3.16: Spherically averaged structure factor for the cylindrical phase in the systemfA= 0.3, χN = 65 and the chain lengthN = 100. The positions of the main peakk1and two higher order peaks k2 andk3 are marked with 1, 31/2 and 71/2 respectively with their relative positionsk2/k1= 31/2 and k3/k1= 71/2.
3.5 Asymmetric Chains
Figure 3.17: Isosurfaces of the cylindrical phase in the system as in Fig. 3.16. The columnar tubes of the minority component are ordered on a two-dimensional hexagonal lattice. Connections between tubes are in fact additional domains of cylindrical structrure with different orientations.
Figure 3.18: Isosurfaces of the bcc phase in the system of asymmetric chainsfA= 0.17 atχN = 45.0 (chain length N = 150). The minority component is contained in spheres ordered in a bcc lattice.
4 Dynamic Properties in the Bulk
Now we discuss how the self-diffusion of diblock copolymers is influenced by concentration fluctuations when approaching the ODT from the disordered phase. Another issue is the effect of the ordered structures on the diffusion. As the GDM does not catch molecular features on the length scale less than the gyration radius of the block, we do not intend to study the diblock copolymer dynamics on the length scales less than the gyration radius of the molecule RG and the time scale less than the disentanglement time τD, see section 1.
For all systems the time dependent directionally averaged diffusion coefficient is defined from the mean square displacement of the center of mass of the molecule
D(t) = 1 3
[~rcm(t)−~rcm(0)]2
6t , (4.1)
where h...i means average over all molecules and different initial conditions of the system.
4.1 Diffusion in the Lamellar Phase
We will be mostly interested in systems with NA = NB = 60 for which the ODT is ex-pected to occur at χ ∼ 15.0−16.5, section 3.1 of section 3 The mean square displacement [~rcm(t)−~rcm(0)]2
shows usual diffusive behavior for times t > τD for various values ofχ ,see Fig. 4.1. The entanglement time τD can roughly be estimated as an average time which the molecule needs to diffuse over the radius of gyration of one blockRX,which is close to 3.8 for the given chain length.
Close examination of Fig. 4.1 reveals thatτD is of the order 50 MCS and is almost unaffected by the interaction χ , e.g. by the ordering. The short time diffusion coefficient defined as D0 = limt→0D(t) remains constant for all χ , and so is fully determined by D0(χ = 0). We also find by inspection that for all χ the diffusivity D(t) reaches a constant values which are denoted in the following as D .
To study the effect of the lamellar ordering on the diffusion in more details, lamellae oriented perpendicular to thez axis were prepared through placing molecules parallel to the z and with their centers of mass located at the isosurfaces, for details see section 3.2. The system dimension Lz was set to an integer number of the lamellar spacing calculated from the structure factor, see section 3.1 and the lateral dimensionsLx and Ly were taken from Lx =Ly =p
N M/(Lzρtot), where the number of molecules in the system is M = 4000. The systems were then let to equilibrate after which the diffusion coefficients D||(t), Dz(t) were calculated that correspond to the motion in the lateral direction along the lamellae
Dk(t) = 1 2
D
~rcm,||(t)−~rcm,||(0)2E
6t ,
0
Figure 4.1: Mean square displacement of the center of mass of the molecule in disordered and ordered lamellar phases of symmetric chain with NA=NB = 60 at variousχ .
and to the motion across the lamellae D⊥(t) =
[z(t)−z(0)]2
6t ,
where ~rcm,|| = (x, y). The quantity D
~rcm,||(t)−~rcm,||(0)2E
is plotted in Fig. 4.2; it shows no suppression of the lateral diffusion, instead a slight increase in the lateral motion of the molecules is observed with increasing χ, which may be attributed to the reduction of the structural fluctuations and so to an decrease of the A−B interface thickness leading to more free motion of the molecules along the isosurfaces. The overall weak dependence of the mean-square displacement on χagrees with the prediction of Barrat and Fredrickson [Bar91] that in a system of Rouse chains the diffusion of the molecules along lamellae is not affected by the parameter χ .
Contrary to the lateral behavior, the diffusion of the center of mass of the molecules in the direction perpendicular to the lamellar layers is suppresed when increasing the interaction χ , see Fig. 4.3. For χ ≥ 30.0, the quantity
[z(t)−z(0)]2
practically saturates at a constant value, meaning that the centers of mass of the molecules are almost attached to the isosurfaces with single jumps from one isosurface to a neighbouring one.
All the measured diffusion constants normalized byD(χ= 0) are summarized in Fig. 4.4. We find a continuous drop in the isotropic diffusion coefficient Dstarting already in the disordered melt. That means that the A and B−blocks of the molecule prefer to stay in A and B-rich regions already forming in the disordered phase. Near the ODT, the quantity D is about 80%
of its value atχ= 0 which is higher than the theoretical prediction of about 60% by Barrat and
4.1 Diffusion in the Lamellar Phase
0 5 10 15 20 25 30 35 40
0 100 200 300 400 500
0.5 〈[ r cm,|| (t)-r cm,|| (0) ] 2 〉
t
χN= 15.0 20.0 16.5 30.0 54.0
Figure 4.2: Lateral mean square displacement of the center of mass of the molecules in the ordered phase of symmetric chain withNA=NB= 60 at variousχ .
0 5 10 15 20 25 30 35
0 100 200 300 400 500
〈[ r cm,z (t)-r cm,z (0) ] 2 〉
t
χ N= 15.0 16.5 20.0 30.0 54.0
Figure 4.3: Perpendicular mean squared displacement of the center of mass of the molecules in the ordered phase of symmetric chain withNA=NB= 60 at variousχ .
0 0.2 0.4 0.6 0.8 1 1.2 1.4
0 10 20 30 40 50 60
D/D( χ =0)
χ N
disordered phase
ordered phase disordered
phase
ordered phase disordered
phase
ordered phase
D D ||
D ⊥ Equation
Figure 4.4: Normalized diffusion coefficient Das well as anisotropic diffusion coefficientsDk, D⊥ in the lamellar phase versus χN in the system of symmetric chains withN = 120. The vertical dashed dotted line separates isotropic from anisotropic diffusion. Black triangles correspond to diffusion coefficient calculated within the mean field approach, see Eq. (4.2).
Fredrickson [Bar91] supported by simulation results of Blumen [Hof97a] and Kremer [Mur99]
who observed a drop of about 40% in the isotropic diffusion near the ODT. D is consistent with the average D≈(2Dk+D⊥)/3≈(2/3)Dk in the lamellar phase. It continues to decrease further and at χ ≃ 25.0 saturates reaching about 60% of its value at χ = 0 characteristic of two-dimensional diffusion, see Fig. 4.4.