Characterization of the dynamics of block copolymer microdomains with local morphological measures

We investigate the structure formation in thin films of cylinder forming block copolymers. With in situ scanning probe microscopy image sequences can be recorded with high temporal (2 min per frame) and spatial (10 nm) resolution. We compare different image processing methods for quantitative analysis of the large amount of data. Computing local Minkowski functionals yields local geometrical and morphological information about the observed structures and enables us to track their evolution with time. An alternative characterization method is to reduce the gray scale images to their skeleton and to classify and count the branching points of the skeletonized structure. We tracked the temporal evolution of these measures and computed correlation functions.

Figure 1

(a) Tapping mode SPM phase image $P\left(x,y\right)$ of a thin film of a cylinder forming polystyrene-block-polybutadiene-block-polystyrene triblock copolymer (data set A). Bright corresponds to polystyrene microdomains. Depending on the film thickness the block copolymer shows different structures, such as parallel cylinders (bright stripes), upright standing cylinders (bright dots), or perforated lamellae (dark dots). Here, the film thickness decreases from left to right. (b) Map of the local Euler characteristic $X_{39}\left(x,y\right)$ of (a) calculated with box size $14\times 14\text{pixels}$ and threshold $\rho =0.39$. It can be used as a morphology map mapping different structures to different gray values. All quantities are dimensionless.

Figure 2

Area $A$, Euler characteristic $X$, perimeter $P$, and the ratio $X/P$ as functions of the threshold $\rho$ calculated for different samples with different morphologies using a box size of $14\times 14\text{pixels}$. The first row shows examples of the different types of microdomain structure. In each diagram four curves are shown which correspond to four different samples of the same type of structure. All quantities are dimensionless.

Figure 3

Tapping mode SPM phase image of a polystyrene-block-polybutadiene diblock copolymer (image size $213\times 221$) swollen in chloroform vapor (data set B). The image is the first of a sequence of images displaying the microdomain dynamics during further annealing. The temporal evolution of region (a) and (b) is shown in Fig. 4. Crosses and numbers indicate the position of samples that are analyzed in Figs. 7, 8.

Figure 4

Two samples ($40\times 40\text{pixel}$) of the film shown in Fig. 3 observed during increasing saturation with chloroform vapor from 10% to 80%. The individual images show the dynamics of the film. The frame rate is about 2 min per frame. The numbers indicate the frame numbers. (a) Position (a) of Fig. 3. The whole data set consists of 161 images. (b) Position (b) of Fig. 3. The whole data set consists of 127 images.

Figure 5

Gray scale maps of the Euler characteristic $X$ at box size $40\times 40$ as a function of density threshold $\rho$ and frame number $n$: (a) of the sample shown in Fig. 4a, (b) of the sample shown in Fig. 4b. All quantities are dimensionless.

Figure 6

Euler characteristic $X_{39}\left(n\right)$ and $X_{33}\left(n\right)$ for density threshold $\rho =39$ and $\rho =33$, respectively, versus frame number $n$: (a) corresponds to the sample shown in Fig. 4a, (b) corresponds to the sample shown in Fig. 4b. All quantities are dimensionless.

Figure 7

Correlation function of the Euler characteristic $X_{39}\left(n\right)$ versus time lag $\tau$. The five curves correspond to different samples of Fig. 3, where sample 3 corresponds to Fig. 4a and sample 4 corresponds to Fig. 4b. (a) Correlation functions calculated in the intervals [1,161] and [1,127], (b) correlation functions calculated in the interval [1,100]. All quantities are dimensionless.

Figure 8

Averaged pixel-to-pixel correlations of image samples $P\left(x,y\right)$ (where $x$ and $y$ are the spatial coordinates) versus time lag $\tau$. The curves correspond to the same samples as in Fig. 7. At three spots error bars indicating the standard deviation of the correlation function corresponding to sample 3 are shown. The standard deviation is similar for all $\tau$ and for all samples. All quantities are dimensionless.

Figure 9

Reduced skeleton of frame 1, 67, and 161 of the sequence of images shown in Fig. 4a.

Figure 10

(a) Number of $y$ connections versus frame number $n$ of samples 3 and 4 corresponding to the data sets shown in Fig. 4a and Fig. 4b, respectively. (b) Correlation function of the number of $y$ connections versus time lag $\tau$. Sample 3 and 4 correspond to the data sets shown in Fig. 4a and Fig. 4b, respectively. For sample 3, also the correlation function for frame 1 to frame 100 is shown. All quantities are dimensionless.