PART I
THE MASK
CHAPTER II
Development of an image analysis tool for
colloidal mask assessment
The present chapter is focused on the tuning of an image analysis tool (Matlab) that will be used for the quantitative assessment of the quality of colloidal masks used in nanosphere lithography. The outline of the procedure is discussed and illustrated by practical examples.
Finally, the reliability of the program is
evidenced.
CHAPTER II -‐ Development of an image analysis tool for colloidal mask assessment -‐-‐-‐-‐-‐-‐-‐-‐ 35 1. Introduction -‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐ 37 2. Experimental part -‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐ 38 3. Results and discussion -‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐ 38 3.1 Image acquisition -‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐ 38 3.2 Image preprocessing -‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐ 39 3.2.1 Preamble -‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐ 39 3.2.2 Preprocessing steps -‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐ 41
Removal of background -‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐ 43
Erosion -‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐ 44
Contrast enhancement -‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐ 46
From segmentation to binarization -‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐ 47 3.3 Image analysis -‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐ 48 3.3.1 Compaction rate/surface coverage -‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐ 49 3.3.2 Number of hexacoordinated spheres -‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐ 52
Voronoï diagrams -‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐ 52
Calculation of distances -‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐ 54
3.4 Image interpretation -‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐ 54
4. Conclusions -‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐ 57
5. References -‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐ 58
-‐ 37 -‐
1. Introduction
During the last decade, nanoparticles and nanostructures composed of metals or semiconductors have called a lot of attention due to their new optical, chemical, electrochemical, catalytic, or biological characteristics, which are not obtainable with conventional bulk materials.
Conventional nanolithographic techniques (e.g. electron beam lithography) are frequently used to pattern solid surfaces with regular arrays of nanometer scale features. However, besides some practical limitations, only few laboratories can afford these methods that are cost-‐consuming. Alternatively, many research activities are currently devoted to less expensive and parallel nanoscale methods for regular structure formation. Nanosphere lithography is one of these methods. First, a colloidal crystal
[1]mask (CCM) is obtained from a self-‐organized monolayer of nanobeads on a substrate. There are several methods for the formation of the self-‐organized particles monolayers such as the Langmuir-‐Blodgett technique, electrostatic deposition, controlled evaporation of solvent from a suspension containing latex particles, floating-‐
transferring technique or spin-‐coating (Chapter I -‐ Section 2.1 Designing monolayers).
Due to the rather small size of most polymer latex particles, investigation of the morphology of aggregated colloidal particles has not always been easy.
Studies
[2-‐4]have been reported on the aggregation/rearrangement of macroscopic spheres. Nevertheless, various characterization parameters, such as Fourier transform analysis,
[5]diffraction properties
[6]or average size of ordered domains,
[6, 7]have been used to evaluate the quality of colloidal masks. However, these analysis methods are insufficient to quantify the presence of irregularities in the monolayers.
Moreover, most researchers give a rough idea of the size of the monolayers areas, ignoring most of the time the presence of small defects and usually emphasizing on a perfect structure rather than on deviations. Knowing the nature and the precise amount of the defects, however, one can try to reduce them by avoiding the conditions for their formation. Therefore, it is important to study the microscopic structure of the colloidal mask in order to correlate the quality of the sample to its growth methods and conditions.
One major goal is to develop an image-‐processing tool to evaluate the quality of such masks. The prime objective of an image processing
[8]system is to reduce a huge amount of apparently random data (cleaning parasite information), into a few deciding features to highlight the interesting information and extract judicious data from an image.
Secondly, the time saving due to computerized analysis is a definite advantage. This
program will be used (chapter III) to investigate the impact of selected spin coating
parameters on the presence of defects.
2. Experimental part
Monodisperse polystyrene (PS) nanospheres with a mean diameter of 490 nm were purchased from Bangs Laboratory as suspensions in water (concentration of about 10 % wt). PS colloidal masks were synthesized by a spincoating process on polished quartz substrates. The preparation of these masks will be further detailed in chapter III.
Samples were then submitted to image processing.
Scanning electron microscopy (SEM) analyzes were performed on a FEG-‐ESEM XL30 (FEI) with an accelerating voltage of 15 kV and a back-‐scattered electron (BSE) detector under high vacuum. All samples were gold-‐coated (60 s) before observation. Each micrograph (8-‐bit image) is composed of 484 x 712 pixels and was acquired at 4000 x magnification. The grey level then varies from 0 (pure black) to 255 (true white).
Image preprocessing and image analysis were implemented with a computer program (Matlab , 7.1 version), together with its image processing toolbox. Interpretation of the results provided by the program was performed by casting a critical eye on them.
3. Results and discussion
Image processing is usually subdivided into four major steps :
• Image acquisition;
• Image preprocessing;
• Image analysis;
• Image interpretation.
The ins and outs of all these steps are discussed below.
3.1 Image acquisition
This first step is critical and requires special attention because the quality of the starting images is largely responsible for the smooth running of the sequence of operations. If the image has not been acquired satisfactorily then the intended tasks may not be achievable, even with the help of some form of image enhancement.
Information that can be drawn from those micrographs depends crucially on the magnification at which they were made. To give an idea of surface coverage, rather low magnification micrographs are sufficient (Figure II -‐ 1 (a)) while higher magnifications are required to discriminate each nanosphere (Figure II -‐ 1 (b)).
-‐ 39 -‐
Figure II -‐ 1
SEM micrographs of a self-‐organized PS nanospheres (490 nm diameter) with multiple defects.
(a) 1000 x magnification (484 x 712 pixels). The inset (scale bar is 5 µm) is a zoom on a 50 x 50 pixels area.
(b) 4000 x magnification (484 x712 pixels). The inset (scale bar is 1 µm) is a zoom on a 50 x 50 pixels area.
Moreover, image acquisition should be performed as much as possible in the same contrast and brightness conditions in order to facilitate the analysis.
3.2 Image preprocessing
3.2.1 Preamble
Image analysis is by definition a quantitative measure. However, as-‐acquired micrographs can seldom be used to extract pertinent information. Several treatments of images are usually required to optimize the quantifications.
The aim of preprocessing is to improve the image data in order to suppress undesired
distortions or to enhance some image features (segmentation) relevant for further
processing and analysis task. In addition, for the sake of automation, the treatments
should be applied to an entire set of images so that the measurements are not based on individual adjustment of these images. The ultimate goal is to obtain a binary image in which it is possible to discriminate between different nanospheres, white and black colors coding respectively for the spheres and the voids.
The first step in image preprocessing is image cropping. Some irrelevant parts of the image can be removed to focus on the region of interest. Micrographs were cropped to a 423 x 712 pixels zone to remove the data bar (Figure II -‐ 2(a)).
Before going deeper into details, it should be mentioned that preliminary preprocessing tests were performed using the software LUCIA . However, even if the program allows a semi-‐automated analysis, several defects/cracks were not detected (Figure II -‐ 2) and it therefore required the use of a stylus for manual detection. This increases consistently the analysis time and adds a human factor, which may be an additional source of error in the process (non-‐detection of certain defects, etc.).
The use of Matlab therefore appeared to be obvious, because of its applicability and versatility.
[9]Figure II -‐ 2
(a) SEM micrograph of a self-‐organized PS nanospheres monolayer (490 nm diameter). Scale bar is 10 µ m.
(b) Same micrograph during preprocessing with LUCIA . The green color is supposed to highlight the nanospheres areas. Blue and yellow arrows point respectively towards wrongly and undetected defects.
!"#$
!%#$
-‐ 41 -‐
3.2.2 Preprocessing steps
Figure II -‐ 3 illustrates the pixel values in the various regions of a polystyrene nanospheres monolayer micrograph (Figure II -‐ 3 (a)).
A nanosphere (Figure II -‐ 3 (b)) is composed of light-‐gray pixels and is surrounded by six darker spots attributed to the interstices between the nanospheres.
In the small defects areas, such as vacancies (Figure II -‐ 3 (c)), the pixels present a dark gray contrast. On the opposite, the large non-‐covered areas present pixel values very close to those inside the nanospheres.
This may be a major problem to resolve foreground from background (segmentation).
Usually, this segmentation process is based on the image gray-‐level histogram, which represents the distribution of the pixels in the image over the gray-‐level scale
*. In that case, the aim is to find a critical value or threshold. Through this threshold, applied to the whole image, pixels whose gray levels exceed this critical value are assigned to one set and the rest to the other, i.e. respectively nanospheres and background in our case.
For a well-‐defined image, its histogram should have a deep valley between two peaks, which unfortunately is rarely the case due to a number of conditions like poor image contrast or spatial nonuniformities in background. These cases require user interaction for specifying the desired object and its distinguishing intensity features, through use of mathematical morphology.
Mathematical morphology is a set theory approach developed by Matheron
[10]and Serra.
[11]It provides an approach to digital image processing based on geometrical shape. Indeed, the basic idea of mathematical morphology is to compare the items you want to analyze with another object of known shape (e.g. disk, square, line etc.) called structuring element. The fundamental morphological operators are the dilatation and the erosion (Appendix A); one of the simplest applications of the dilatation is bridging gaps, whereas the erosion deletes irrelevant details. For more information, readers should consult the latest Matlab Help File
[12]available online.
*
As a reminder, the values are on a scale of 0 to 255, whereby 0 corresponds to black and 255
corresponds to white.
Fi gu re II -‐ 3 Pi xe l v al ue s in v ar io us a re as o f a s el f-‐ or gan iz ed PS m on ol aye r. (a ) SE M m ic rogr ap h of a se lf-‐ or gan iz ed PS m on ol aye r. Sc al e bar is 4 µ m. (b ) Pi xe l v al ue s in a 13 x 13 pi xe ls a re a fo cu se d on a n an os ph er e. (c ) Pi xe l v al ue s in a 13 x 13 pi xe ls a re a fo cus ed on va ca nc y ar ea . (d ) Pix el v al ue s in a 1 3 x 13 p ix el s are a fo cu se d on a l arg e no n-‐ co ve re d ar ea .
!" #$ !%#$ !& #$ !'#$
-‐ 43 -‐
The main preprocessing steps are illustrated on the basis of a polystyrene monolayer mask (Figure II -‐ 4), and chronologically represented from Figure II -‐ 4 to Figure II -‐ 12.
The MatLab script file is presented in Appendix B.
The gray-‐level histogram in Figure II -‐ 4 presents a broad peak in the light-‐gray range attributed to the nanospheres and to the large non-‐covered areas.
Figure II -‐ 4
SEM micrograph of a PS (490 diameter) monolayer and its corresponding gray-‐level histogram.
Removal of background
As previously illustrated, the background illumination is brighter in large non-‐covered areas than in small defects areas or in the interstices between the nanospheres.
In order to identify the nanospheres from the background, we first need to estimate the background illumination. That was performed by a morphological opening operation.
Morphological opening consists in an erosion step followed by a dilatation step, using the same structuring element for both operations. It has the effect of removing objects that cannot completely contain the structuring element.
In our case, we need to remove the nanospheres from the image and the structuring element must be sized so that it cannot fit entirely inside a nanosphere. According to Figure II -‐ 3 (c), the diameter of a nanosphere is around 9 pixels. We tested a disk-‐
shaped structuring element with various sizes.
The obtained background was then subtracted from the original image.As expected, a too small (resp. too large) structuring element over-‐ (resp. under-‐) estimated the background (Figure II -‐ 5). The optimum size was found to be 10 pixels (Figure II -‐ 6).
Figure II -‐ 5
Zoom on background-‐subtracted images obtained with disk-‐shaped structuring elements of two different sizes.
!"#$%&'%
!"#$%(%
Figure II -‐ 6
Optimum background-‐subtracted image with a disk-‐shaped structuring element of size 10 and its corresponding gray-‐level histogram.
The comparison between the gray-‐level histograms in Figure II -‐ 4 & Figure II -‐ 6 clearly shows the separation of the peak (left-‐sided) corresponding to the background.
The width of the peak corresponding to the nanospheres is attributed to the fact that it includes pixels located at the contact points between the nanospheres (Figure II -‐ 3 (b)).
The binarization of this background-‐subtracted image (Figure II -‐ 6), rather than to give a representation of the organization of the nanospheres, shows a foretaste of the nano-‐
objects that could be formed using this mask (Figure II -‐ 7).
Figure II -‐ 7
Binarized image of figure II -‐ 6. The inset highlights the expected triangular nanodots that could be manufactured using the polystyrene monolayer as a mask.
Erosion
An erosion step was therefore performed on Figure II -‐ 6 with a disk shaped structural element. The impact of disk size on the nanosphere shape is evidenced in Figure II -‐ 8. A disk of size 5 erodes too aggressively the spheres, while bridges between the beads are still remaining with a disk of size 1. Optimum size of 3 was then chosen.
!"#$%&'%
-‐ 45 -‐
By the way, it has to be mentioned that the use of O
2-‐plasma etching will be broached in chapter III. This process may considerably change the starting picture by proceeding to
“physical” erosion of the spheres instead of “computerized” erosion of which we have been discussing here above. This etching step isolates the spheres from each other and may facilitate their discrimination. Moreover, it is useful to note that our program was also successfully tested on multilayer samples. This particular issue will be discussed in section 3.3.1.
Figure II -‐ 8
Erosion with a disk shaped structuring element of size 1, size 3 and size 5 and their corresponding gray-‐
level histograms. Insets are zooms on 50 x 50 square pixels area.
!"#$%&%
!"#$%'%
!"#$%(%
Contrast enhancement
Looking back in Figure II -‐ 8, it is obvious that erosion affected the brightness of objects, which is reflected by a lack of contrast against the background. A common technique for contrast enhancement is the combined use of the top-‐hat and bottom-‐hat filtering.
The top-‐hat transform (Matlab function imtophat) is defined as the difference between the original image and its opening. The opening of an image is the collection of foreground parts of an image that fit a particular structuring element.
[12]The bottom-‐hat transform (Matlab function imbothat) is defined as the difference between the closing of the original image and the original image. The closing of an image is the collection of background parts of an image that fit a particular structuring element.
[12]Top-‐hat filtering is used to intensify valleys in an image, while bottom-‐hat filtering enhances contrast.
Since the objects of interest in our image are the nanopsheres, we again used disk-‐
shaped structuring elements. The size of the disk is based on an estimation of the average radius of the objects in the image and a size of 3 was chosen (Figure II -‐ 9).
Figure II -‐ 9
Enhanced contrast image of Figure II -‐ 8 (size 3) and its corresponding gray-‐level histogram.
The top-‐hat image contains the "peaks" of objects that fit the structuring element. The imbothat function shows the gaps between the objects. To maximize the contrast between the objects and the gaps that separate them from each other, we added the top-‐
hat image to the original image (Matlab function imadd), and then subtracted the
"bottom-‐hat" image from the result (Matlab function imsubtract).
Compared with Figure II -‐ 8 (size 3), the gray-‐level histogram in Figure II -‐ 9 is now composed of two diametrically opposite sharp peaks, which is the ultimate goal of segmentation.
-‐ 47 -‐
From segmentation to binarization
Following the segmentation is the binarization, which refers to the conversion of the gray-‐level image (Figure II -‐ 9) into a binary image (Figure II -‐ 10). Binarization can be achieved by thresholding, i.e. by assigning all the pixels with gray-‐level lower than a given threshold to either the background or the foreground, and the remaining pixels to the other set. The threshold was set with the function graythresh.
The graythresh function uses Otsu's method,
[13]which chooses the threshold so that the variance of the pixels values within each class are minimized. The Otsu method still remains one of the most referenced thresholding methods. However, a manual thresholding is always possible in case of problems.
The binarized image is presented in Figure II -‐ 10. The pixels are now distributed between black (0) and white (1) values. Unfortunately, the inset in the binarized image shows the presence of bridges between the nanospheres, which is a major problem for the discrimination of the different entities. To remove the bridges between the nanospheres, we performed an opening treatment with a disk shaped structural element of size 1 (Figure II -‐ 11).
Figure II -‐ 10
Binarized image of Figure II -‐ 9 and its corresponding distribution of black and white pixels, The inset highlights some bridges between the nanospheres.
Figure II -‐ 11
Binary image of Figure II -‐ 10 after opening operations to remove bridges and its corresponding
distribution of black and white pixels.
As the amount of bridges is not so large, no drastic change is observed in the binary level histogram. However, the physical separation of the balls is crucial to the subsequent image analysis.
Finally, a last type of noise pollution, known as impulsive noise, can disturb image analysis process. Impulsive noise, also called “salt and pepper” noise, is a degradation of the image where some pixels randomly become either white or black. It is generally assumed that the probability that a pixel is white (resp. black) is constant on the image, and the fate of each pixel is independent from the others.
Removal of this kind of noise was performed through the application of a median filter.
What a median filter does is to take all the pixels within a certain radius around a pixel, sort them numerically and then return the color that ends up in the middle of the sorted list.
Now that the preprocessing was successfully carried out, binary images can be submitted to image analysis.
Figure II -‐ 12 illustrates an example of black and salt pepper noise and its removal.
Figure II -‐ 12
(a) SEM micrograph of self-‐organized PS nanospheres (490 nm diameter) monolayer.
(b) Binarized image wit salt & pepper noise.
(c) Binarized image after removal of salt & pepper noise with a median filter of size 3.
3.3 Image analysis
The third image-‐processing step is the extraction of meaningful information quantifying a phenomenon.
In our case, the aim of setting up such an image analysis tool is to assess the quality of colloidal crystal masks. Several strategies, based on different concepts, can be considered. Each of them are discussed below taking into account their advantages and drawbacks.
(a) (b) (c)
-‐ 49 -‐
3.3.1 Compaction rate/surface coverage
The first method that comes to mind is to measure the proportion of area covered by the nanospheres, commonly known as the compaction rate. It is well known that hexagonal close packing is the most efficient way to pack spherical particles in 2D. The inset below shows the calculation of the rate of hexagonal close-‐packed (HCP) compaction rate.
In case of monolayers, the best 2D packing is reached, as the coverage rate gets closer to 90 percent. The detailed calculations appear in the box below.
The presence of defects such as dislocations or vacancies in the monolayers should inevitably lead to the drop of the compaction rate.
On the other hand, multilayer samples should also perturb the HCP rate, depending on the way the additional layers arrange.
The tuning of surfactant concentration as well as the presence of multilayer samples are some factors that can induce slight modifications (in terms of illumination/contrast) in the starting micrographs and hence affect the final quantified value.
Various examples are represented in Figure II -‐ 13 (a), (c) and (e).
In case of high surfactant concentration, a dark gray ring surrounds the nanospheres (inset of Figure II -‐ 13 (c)) and isolates them from each other. The use of an erosion step is therefore not required.
!"#$%&'()%*+&,$-)*.$$
!"#$%&'%()#%*+,(%-#..%/%
2 4d! 3 60 sin 2d 2d h
2d ! = ! = %
0*12#"%&'%3$"(,-.#4%,+%()#%*+,(%-#..%/%56 4 ! 1 2
"
# $ %
&
'+ 2 ! 1
3
"
# $ %
&
' + 2 ! 1
6
"
# $ %
&
' /%7%%
!"#$%-&8#"#9%2:%()#%3$"(,-.#4%/%
2 22
4 ! d " = ! d
#
% $
&
' %
!%;<=%-&13$-(,&+%"$(#%%%/% ! d
24 d
23
2
= !
2 3 ! 0, 9 %
!
!
!
!
!
!
!
!
!!"! !
!
!
!
!
!
!
!
d
#$%&!'())!
Fi gu re II -‐ 13 M m ic rogr ap hs an d cor re sp on di ng bi nar y im age s of var iou s se lf-‐ or gan iz ed P S m on ol aye rs ( 490 nm di am et er ). Sc al e bar s ar e 5 µ m. T he in se ts a re a z oo m in 4 µ m ar ea . ) & (b ) Re sp ec tiv el y, S EM m ic ro gr ap h an d bin ar y of a se lf-‐ or gan iz ed PS m on ol aye r & (d ) Re sp ec ti ve ly , S EM m ic ro gr ap h an d bi na ry im ag e of a se lf-‐ or gan iz ed PS m on ol aye r at hi gh su rf ac ran t c on ce nt rat ion ) & (f ) R es pe cti ve ly , S EM m ic ro gra ph a nd b in ary im ag e of a PS mu lt ila ye r
!"#$ !"#$ #$ #$
(f ) (d )
-‐ 51 -‐
In case of multilayer samples (Figure II -‐ 13 (e)), easily recognizable by the detection of underlying nanospheres throughout the defects of the superficial layer, the contrast around each nanosphere may also vary, depending on the packing of the nanospheres.
The packing of the nanospheres in a square lattice will be discussed in chapter III.
However, all these samples were successfully binarized with the help of our homemade program and the corresponding binary images are presented in Figure II -‐ 13 (b), (d) and (f).
The comparison between insets of Figure II -‐ 13 (b), (d) and (f) highlights a visible difference to the naked eye between the size of binarized spheres, which is due to the different gray levels present in the starting micrographs.
The size of the binarized nanospheres is hence highly dependent on the preparing conditions of the samples.
A dilatation step of the binarized particles could be envisaged but due to the small number of pixels between the spheres, the creation of bridges between them is hardly inevitable.
This change in particle size directly affects the rate of coverage by the beads and is clearly highlighted in Figure II -‐ 14, which plots the proportion of white pixels present in Figure II -‐ 13 (b), (d) and (f).
Figure II -‐ 14
Evolution of the % of white pixels in binary images presented in Figure II -‐ 13.
The first comment concerns the numerical value of this coverage rate, which is relatively low. What is more embarrassing is that the coverage rate of the multilayer sample is unfairly the lowest and almost equivalent to the one of the monolayer in Figure II -‐ 13 (a).
Given the apparent higher number of particles in the multilayer sample, we would have indeed expected a higher coverage rate.
For all these reasons and after several tests it was decided not to opt for this criterion to assess the quality of masks.
!"!#
$"!#
%!"!#
%$"!#
&!"!#
&$"!#
'!"!#
()*+,-#../#%'#012#
()*+,-#..#/%'#032#
()*+,-#..#/#%'#042#
5#64#78)9-#:);-<=#
3.3.2 Number of hexacoordinated spheres
The second way to assess the quality of the colloidal masks is to calculate the rate of hexacoordinated particles. The Matlab help file has inspired two different strategies. The first is based on the construction of Voronoï diagrams and the second one on the calculation of distances between particles.
Voronoï diagrams
Even though Voronoï diagrams were first investigated by René Descartes
[14]in the 17
thcentury and applied by Dirichlet
[15]when exploring quadratic forms, the diagrams were named by Georgy Voronoï.
[16]He published a generalization of this concept that would apply to higher dimensions and so introduced the concept in its modern form.
A Voronoï diagram is a special kind of decomposition of a metric space determined by distances to a specified discrete set of objects in the space, e.g., by a discrete set of points. It is called a Voronoï tessellation, a Voronoï decomposition, or a Dirichlet tessellation.
In the simplest case, let us consider a set of points in the plane (Figure II -‐ 15 (a)), which are the Voronoï sites. Voronoï diagrams may also be constructed from a set of points in the 3D space. Each site has a Voronoï cell consisting of all points closer to itself than to any other site (Figure II -‐ 15 (b)). The shape of this cell (or polygon) depends on the number of neighbors. The segments of the Voronoï diagram are all the points in the plane that are equidistant to the two nearest sites.
Figure II -‐ 15
(a) A set of points in the Euclidian space (centers are in dark gray)
(b) Voronoï polygon of the central disk. The red lines are the perpendicular bisectors to the black lines connecting the centers of two neighboring disks.
The use of Voronoï constructions has been reported by Marcus
[17]to study phase transitions (liquid <> solid) in a confined quasi-‐two-‐dimensional colloidal suspension.
Our program was developed according the following approach.
First, every single particle was identified and assigned as a centroïd whose center
coordinates were placed in a matrix. The Voronoï diagram was then built up. The rate of
hexacoordinated particles was obtained by counting the number of hexagonal Voronoï
cells related to the total number of particles. The Voronoï diagrams of the previously
-‐ 53 -‐
Figure II -‐ 16
Voronoï diagrams of respectively:
(a) Figure II -‐ 10 (a). The inset presents hexagonal Voronoï cells.
(b) Figure II -‐ 10 (c). The inset presents hexagonal Voronoï cells.
(c) Figure II -‐ 10 (e). The inset presents, among other, square Voronoï cells.
!"#$
!%#$
!&#$
Hexagons are observed among other polygons in the insets. The total number of particles and the proportion of hexacoordinated ones will be discussed in section 3.4, dedicated to the interpretation of results. They will be compared with the results obtained with the next method based on calculation of distances.
Calculation of distances
This process also required the creation of a matrix containing the center coordinates of every single particle. The program computed the Euclidean distance between all the centers in a new matrix.
Since the PS particles are monodisperse, any particle that lay at a lower distance than the diameter of the PS spheres + ε was considered as a close neighbor.
Thanks to the image toolbox of Matlab, we estimated this distance to 12 pixels, which was kept constant for all analysis.
3.4 Image interpretation
At last, one of the most important stages of image processing is probably the interpretation of the results. Indeed, it is essential to keep a critical eye on each step of the image processing to ensure that no distortion of images vitiates the outcome.
Manual counting was used to validate the accuracy of the program.
The total number of particles evaluated by manual or computerized counting in Figure II -‐ 13 (a), (c) and (e) are plotted in Figure II -‐ 17.
The percentage of error in all three cases is less than 1 %, thereby validating the counting method.
Knowing the dimensions of the micrographs, the coverage rate can be inferred with greater acuity than with the method measuring the proportion of white/black pixels (Section 3.3.1 Compaction rate/surface coverage).
!"
#!!"
$!!!"
$#!!"
%!!!"
%#!!"
&!!!"
&#!!"
!"#$%&'(()'*+',-.'
!"#$%&'((')*+',/.'
!"#$%&'((')'*+',&.'
0123' 0344' 05*2'
0+51' 0330' 0526'
!"#$%&'()*+,&"-&.$,/0%+1&
7-8$-9':;$8<"8#' :;=>$<&%"?&@':;$8<"8#'
-‐ 55 -‐
Regarding the number of hexacoordinated particles, several remarks can be made at the sight of Figure II -‐ 18.
First, the number of hexacoordinated particles provided by the program based on Voronoï diagrams is, for each sample, higher than those provided by manual counting and the program of distances.
Figure II -‐ 18
Number of hexacoordinated particles in Figure II -‐ 10 (a)/(c)/(e) by manual and computerized counting.
This difference is particularly glaring (deviation of more than 300 %) in the case of the multilayer sample. An explanation must lie in the Voronoï diagram itself.
At first glance, the proportion of hexagons in Figure II -‐ 16 (c) appears to be very low.
However, if we take a look at higher magnification, the square-‐shaped Voronoï cells believed to account for a quadratic arrangement appear to be distorted hexagons (Figure II -‐ 19). Distorted hexagonal Voronoï cells are also found in the case of particles bordering lacunar areas.
The Voronoï program was therefore rejected in favor of the “distance” program, which provides an average deviation rate below 3%. One way to fix the Voronoï program would be to calculate the internal angles of all the hexagons that are numbered in order to exclude the distorted ones.
Another way would be to implement a method based on Delaunay triangulation
[18]procedure, which is dual to Voronoï’s method. The Delaunay triangulation (Figure II -‐
20) is built up by drawing a line segment between any two sites whose Voronoï regions share an edge. A triangle lattice is generated and connects all the particles. The triangles will be equilateral in case of hexagonal close packing.
!"
#!!"
$!!!"
$#!!"
%!!!"
%#!!"
&!!!"
!"#$%&'(()'*+',-.'
!"#$%&'((')*+',/.'
!"#$%&'((')'*+',&.'
*01+' *123' 4*5'
6*+2' 675*' 6*51'
*361' *33*' 4*1'
!"#$%&&'()*$+"(,-$'.%/"0,
1$*2$/,3&2*.*4, 3&5-2+"')6"(,3&2*.*4,, 7898:8;,
3&5-2+"')6"(,3&2*.*4,,
<=>?@:3A,
