Multifractal characterization of unworn hydrogel contact lens surfaces.
Fractal and multifractal theories are efficient tools to better describe biomaterials' surface geometry used in biomedical applications, based on image analysis methods [1-3].
In the field of polymer science, the morphology of multicomponent polymer systems can be described by multifractal analysis [2, 3].
Surface roughness consists of fine irregularities which result from the inherent action of the production process [4, 5]. To characterize surface roughness, two main approaches, based on different mathematical procedures, are generally used: statistical (the classical descriptive approach)  and fractal description (the modern approach) [6-10].
The main characteristic of a fractal description is that it uses scale-invariant parameters, while a statistical description depends on particular length scales, such as instrument resolution or sample length . Therefore, a fractal description has the potential to considerably simplify the statistical approach, and be a valuable addition to methods based on Euclidian geometry.
The roughness of most engineering surfaces has been proved to exhibit fractal characteristics, possessing a self-similar structure over a range of scales. Conversely, it is important to describe rough surfaces by intrinsic parameters that are independent of all scales of roughness, such as the fractal parameters [2, 3].
It is known that polymers possess multilevel different scale structures (molecular, topological, supermolecular, floccular, or block levels), elements of which are interconnected. Polymers possess only statistical self-similarity, which takes place only in a restricted range of spatial scales, from several angstroms up to several tens of angstroms ,
A contact lens is a medical device placed on the cornea of the eye to correct vision, as well as for cosmetic or therapeutic reasons . Each biomedical surface of a CL (inner and outer) is designed to have a specific surface topography that is generated by the manufacturing process and the properties of the biomaterial [12-16], but this is difficult to measure and quantify [11, 17].
CL surface roughness plays an essential role in understanding the tribological interactions of the back surface of the CL with the corneal surface, and the front surface of the CL with the under-surface of the eyelid (friction, wear and elastohydrodynamic lubrication at a nanometer scale) [4, 18-21],
CL roughness maps show local variations in surface roughness values and provide a detailed image of surface heterogeneity and the scale variation of the roughness between different regions/samples.
A variety of microscopy/spectroscopy techniques have been developed in recent decades for the description and evaluation of CL surface characteristics, and these have helped scientific and clinical understanding of the interactions between CL biomaterials and the ocular environment [22-28].
To avoid a number of potential risks (such as irritation to the eye and eyelid surfaces) and corneal complications (such as biological infections) [29-34] many studies have been dedicated to the design and analysis of biocompatible materials [35-42],
Several studies have highlighted the difficulties of CL manufacture and subsequent surface treatment on a commercially viable scale, at a reasonable economic cost, and with adequate quality assurance controls [15, 16],
A smooth surface can reduce the scattered light and improve CL optical performance.
Some studies have demonstrated that the accumulation and deposition of tear components on CL surfaces lead to an increase in surface roughness, depending on the lens type, wearing conditions, and site of the lens , A very rough surface can lead to mechanical irritation on the corneal surface .
Over the last few decades, AFM and fractal analysis have been used extensively in the design and analysis of CL surface topography [17, 24, 25, 27]. The main advantage of AFM is its ability to obtain topographic information from CL surfaces in aqueous, nonaqueous or dry environments, with minimal sample preparation.
The aim of this work is to characterize the surface roughness of unworn hydrogel CL, using AFM and multifractal geometry. To our knowledge, this is the first multifractal analytical study of CL surface roughness in biomaterials literature.
Fractal theory, proposed by Benoit Mandelbrot (1983), has provided a more appropriate and convenient approach to the study of complex and heterogeneous geometrical structures than traditional Euclidean geometry. Fractal theory is suited to the description of objects which have an integer dimensionality and which are ideal, man-made, or regular .
A fractal object is a mathematical set with a high degree of geometrical complexity, mainly characterized by four properties: (a) irregularity of the shape; (b) self-similarity of the structure; (c) noninteger dimension; (d) complexity [6-8].
One of the many different methods of fractal analysis (e.g., the chord method, Minkowski's method, the Slit Island method) is the box-counting method, ideally suited for the analysis of, inter alia, binarized computer images [e.g., two-dimensional (2D) surface profiles] (Fig. 1) [43, 44].
Fractal analysis carried out with this method is very simple. Nets of different mesh sizes (Fig. la) are applied to the binarized image of the surface profile (Fig. 1). For each net size, the meshes in which there is at least a fraction of the studied profile are counted (Fig. lb). The calculation of the slope of the regression line for a double-logarithmic system (Fig. 2) gives the fractal dimension [D.sub.BC], which is described with the following Eq. 1 :
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)
where N(r) represents the number of nonempty boxes of size r needed to cover the fractal structure.
The fractal dimension obtained by this method is in the range of 1 [less than or equal to] [D.sub.BC] [less than or equal to] 2.
Fractal surfaces are continuous but not differentiable. The fractal dimension [D.sub.f] of a surface is a fractional value within the range 2 [less than or equal to] [D.sub.f] [less than or equal to] 3, where [D.sub.f] = 2 (for ideally smooth surfaces, for example, a flat surface) and [D.sub.f] = 3 (for surfaces that occupy all the available volume, for example, an abstract "infinitely rough" surface) [45, 46]. An increasing value of [D.sub.f] represents an increase in surface roughness, a more irregular shape of the spatial structure.
Several studies suggested that the fractal dimension can characterize surface roughness in particular situations, and it can be correlated with some surface roughness parameters [9, 27, 47, 48].
However, the surfaces of multicomponent polymer systems are both geometrically complex and chemically nonuniform, and a simple estimation with a single [D.sub.f] value is insufficient because it does not offer information about the concentration and distribution of asperities with different orders of size [2, 3, 11].
A more detailed characterization of surface roughness in order to reflect the heterogeneity of the surface of multicomponent polymer systems, both locally and globally, can be obtained by applying multifractal analysis.
Multifractal theory generalizes and extends fractal theory from the geometry of sets as such to geometric properties of measure. The multifractal measures (or multifractals) are related to the study of the distribution of different physical quantities in fractal or nonfractal structures [6-8, 49, 50].
A multifractal is a set composed of a multitude of interwoven subsets, each of differing fractal dimensions. The multifractal structure is characterized by a continuous spectrum of fractal dimensions. A multifractal is characterized primarily by a function, while a fractal is characterized primarily by a single number, the fractal dimension. Thus, a fractal may be interpreted as a special case of a multifractal in which all generalized dimensions are equal to the fractal dimension [6-8].
A complex polymer surface is covered by nanoasperities (nanoscale protrusions and cavities having different geometries) with different orders of size. The distribution of nanoasperities may spread over a region in such a way that small nanoasperities appear almost everywhere, medium nanoasperities exist in many places, and large nanoasperities are concentrated only in a few places.
Multifractal measures are related to the study of the distribution of the concentration of asperities with different orders of size on a complex polymer surface.
Using the computer algorithms characterized in papers [45, 51], multifractal analysis was applied by means of the quasi-3D  and 3D [53, 54] methods with reference to the stereometric files of the CL surfaces. The measurements of the surface were made with the highest possible accuracy to meet the criterion of a limit within which the "box" size in the box-counting method approaches zero.
Having information about the number of pixels of a profile's binary image covered by individual boxes, it is possible to determine the probability of finding them in a given box , which constitutes the measure of the analyzed set:
[P.sub.i](r) = [L.sub.i](r)/[L.sub.T](r) (2)
where [L.sub.i]--number of pixels in one box in the given scale r, [L.sub.T]--total number of pixels in all boxes in the given scale r, r--box size in the given scale.
Multifractal measures can be characterized by the generalized fractal dimensions function D(q, r) (also called the fractal spectrum dimensions for a given set) as :
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)
where q is a number from the range -[infinity] to +[infinity]. Furthermore, q represents a selective parameter: high values of q enhance cells with relatively high values for [P.sub.i]; while low values of q favor cells with relatively low values of [P.sub.i]. For the particular case where q = 1, Eq. 3 becomes indeterminate, so D(q, r) is estimated by l'Hopital's rule.
The generalized dimensions, [D.sub.q] for q = 0, q = 1 and q = 2, are known as the capacity (or box-counting), the information (or Shannon) entropy and correlation dimensions, respectively.
[D.sub.0] is the classical fractal dimension that provides average information on the geometric support of the measure. [D.sub.1] quantifies the degree of disorder present in a distribution. [D.sub.2] is mathematically associated with the correlation function and measures the mean distribution density of the statistical measure. [D.sub.q] is a monotonically decreasing function of q.
All dimensions are different, satisfying [D.sub.0] > [D.sub.1] > [D.sub.2]. The limits of the generalized dimension spectrum are [D.sub.-[infinity]] and [D.sub.[infinity]].
Another way of presenting the multifractal measures description is spectrum f([alpha]).
Based on the probability values defined by Eq. 2, a one-parameter family of normalized measures is constructed:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)
where [n.summation over (i=1)] [[[P.sub.i](r)].sup.q] indicates the summation with q exponent order of the probability for all boxes. The fractal dimensions of the subset f[q, r) are determined:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)
indexed with the exponents [alpha](q, r):
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)
The values obtained [alpha](q, r) are called the singularity for the given set. The plot f([alpha]) is called a multifractal spectrum, which presents the whole spectrum of fractal dimensions.
Some of its general properties are as follows: (a) curve f([alpha]) is convex with a single inflection point at the maximum with q = 0; (b) for q = [+ or -] [infinity], the slope is infinite and [[alpha].sub.min] = [D.sub.+[infinity]], [[alpha].sub.max] = [D.sub.-[infinity]]; (c) the spacing of the multifractal spectrum's arms is a measure of the nonhomogeneity of the set analyzed; (d) where the arms' spacing equals zero (i.e., where the analyzed spectrum is reduced to a point), a fractal case takes place, for which one fractal dimension can be determined.
The spectrum arms' height difference is defined as: [DELTA]f = f([[alpha].sub.min]) - f([[alpha].sub.max]). If [DELTA]f < 0 the fragments described by the low probability value predominate; whereas, for [DELTA]f > 0 the fragments described by the high probability value predominate. The left arm of the multifractal spectrum corresponds to strongly irregular areas, defined by a high dimension value; whereas, the right arm of the multifractal spectrum is associated with flat areas, being characteristic for large convex and concave surfaces .
Multifractal analysis of the surface profile from Fig. 1 carried out with the use of the box-counting method gives information about the number of pixels of the profile binary image included in individual boxes (Fig. 3). It enables to determine the probability of finding them in a given box (2), which is the measure of the analyzed set.
A further analysis (3-6) results in the multifractal spectrum, whose maximum at q = 0 is [D.sub.0] = 1.0162, which is exactly the same as the box dimension DBC in fractal analysis (Fig. 2).
A stereometric measurement file can be presented in the form of a contour diagram, a flat grayscale topographic map or color intensity. It can be compared to a grayscale image (Fig. 4).
To present an image as a 3D surface, it is necessary to replace grey points with local maxima (Fig. 5).
Fractal/multifractal analysis of a grayscale image (Fig. 4a) gives the fractal dimension [D.sub.BC] = [D.sub.0] = 2.0. Of course, such an analysis may seem to be pointless, since the dimension will be equal to 2 for each analyzed image. However, it should be emphasized that the multifractal spectrum in multifractal analysis carries relevant information, in particular the distance between its shoulders, which was analyzed in this article.
In our study, the multifractal analyses depend on the experimental and methodological parameters involved in AFM measurements, such as: measurement system, diversity of samples, image acquisition, type of image, image processing, multifractal analysis methods, including the algorithm and specific calculation used, etc. [6-9].
MATERIALS AND METHODS
Five unworn CL samples of vifilcon A, Focus[R] Monthly Toric Visitint[R] model (Ciba Vision Corp., Duluth, GA) were studied .
The Focus[R] Monthly Toric Visitint[R] model has the following properties: specific gravity 1.12; refractive index (hydrated) 1.415; light transmittance 93%; oxygen permeability (Dk) 16.0 x [10.sup.-11] ([cm.sup.2]/s) (ml [O.sub.2]/ml x mm Hg), measured at 23[degrees]C (Fatt method); water content 55% by weight in normal saline. The CL parameters are: chord diameter 14.5 mm; base curves 8.9 mm; refractive power -1.75 diopters (D); light blue visibility tint. The CL material (vifilcon A) is a copolymer of 2hydroxyethyl methacrylate and povidone, USP.
Each sample was extracted and fixed with an adhesive tape onto a special sample holder, designed to mimic the samples' curvature, without inducing material bending.
AFM analysis of the CL anterior optic surface was performed using a commercial Atomic Force Microscope (Veeco D3100, Veeco Instruments) and its associated software . All these experiments were conducted in an aqueous environment using the liquid cell of the AFM with a preservative-free saline solution (0.9% NaCl, Sigma-Aldrich, Germany) to maintain CL hydration during the measurements. The samples were measured at different scanning ranges in order to obtain different magnifications in Tapping Mode[TM].
The experiments were done in the same room, at room temperature (23 [+ or -] 1[degrees]C) and (50 [+ or -] 1%) relative humidity. AFM analysis was performed, on areas ranging from 1 to 4 [micro][m.sup.2], using a Model SNL-10 cantilever (Bruker AFM Probes International, Europe) , a B triangular-shaped silicon nitride cantilever with the following nominal specifications: resonant frequency 23 kHz, spring constant 0.12 N/m, radius 2 nm, length 205 [micro]m and width 40 [micro]m.
The sample was analyzed at a scan rate of 1 Hz with a 512 x 512 pixel image definition on square areas of 4 [micro][m.sup.2] of the anterior optic surface. Scanning resolution of 256 X 256 pixels was applied to the areas 1, 2, 3, and 4 (1 [micro][m.sup.2]). Analyses at two different resolutions were used in order to assess whether the results obtained at the lower resolution will be comparable to those obtained at the higher one. The whole area analyzed by AFM is shown in Fig. 6. The measurements were repeated twice for each sample on different reference areas to validate reproducibility of the observed features (Fig. 7).
For AFM data visualization the free software package Gwyddion 2.28 was used , The multifractal analyses from the AFM data were made using the proposed multifractal analysis method.
Statistical analyses were performed using the Kaleida-Graph software program, version 4.1 (Synergy Software, Reading, PA) , Analysis of variance (ANOVA), followed by a post hoc Tukey's test, was applied to compare, for a given method, if there was a difference in the average [D.sub.q] values between the different square areas. For the areas where differences in the average [D.sub.q] values were found, ANOVA and Tukey's tests were also applied. Differences with a P-value less than 0.05 were considered statistically significant. The average [D.sub.q] results were expressed as mean value and standard deviation.
Multifractal Analysis of CL Surface Roughness
Representative topographic images (2D and 3D) of the anterior optic surface of an unworn vifilcon A contact lens are illustrated in Figs. 8 and 9 (for all scanned square areas). The depth histogram to observe the density of the distribution of the data points on the studied surface is given in Fig. 10.
Details of the volume parameters (surface): Vmp, Vvc, Vmc, and Vvv parameters used to obtain information about the CL surface morphology (Fig. 11) are given in the Appendix.
An analysis of the stereometric files was conducted (Fig. 11) based on the original algorithm (in MATLAB software R2012b, MathWorks), which consists in fractal scaling (in many approximation steps) of the surface measured with an AFM.
For scanning, square areas of 1 X 1 [micro][m.sup.2] were selected every 54th, 36th, 27th, 18th, 12th, 9th, 6th, 4th, 3rd, 2nd, and 1st measuring point of the whole area measured. 11 scalings of the surface were obtained, thus approximating to its real appearance. A change of measured density is a form of surface scaling required during fractal analysis.
For scanning, square areas of 2 X 2 [micro][m.sup.2] were selected every 105th, 70th, 42nd, 35th, 30th, 21st, 15th, 14th, 10th, 7th, 6th, 5th, 3rd, 2nd, and 1st measuring point of the whole area measured. 15 scalings of the surface were obtained, thus approximating to its real appearance. A change of measured density is a form of surface scaling required during fractal analysis.
The multifractal singularity spectrum f([alpha]) for scanning square areas of 1 and 4 [micro][m.sup.2] are illustrated in Fig. 12. The f([alpha]) spectrum was computed in the range -10 [less than or equal to] q [less than or equal to] 10 for successive 1.0 steps.
Tables 1 and 2 present a summary of the f(a) spectrum for scanning square areas of 4 and 1 [micro][m.sup.2] (statistically significant difference: P < 0.05).
Table 3 presents a summary of the generalized dimension [D.sub.q] values, for scanning square areas of 1 and 4 [micro][m.sup.2] (statistically significant difference: P < 0.05).
Representative (2D and 3D) topographic images of the anterior optic surfaces for the CL samples are shown in Figs. 8 and 9, (for scanning square areas of 4 and 1 [micro][m.sup.2]).
The anterior optic surface of all the CL samples is covered by nanoasperities (nanoscaled protrusions and cavities having different geometries and orders of size) and a specific distribution due to the manufacturing processes, which is evident for the entire magnification range. This pattern is a clear indication of the multifractal nature of the real surface.
Calculations were performed for each data set in the range -10 [less than or equal to] q [less than or equal to] 10 in q steps of 1.0. In all of the calculations [D.sub.0] > [D.sub.1] > [D.sub.2], indicating that the CL surface roughness had a tendency toward the multifractal scaling property. A statistically significant difference (P < 0.05) was found for all [D.sub.q] values. Results of the f([alpha]) spectrum and the generalized dimensions [D.sub.q] values are summarized in Tables 2 and 3.
In the box method described with formulas 2-6, the fractal dimension of the surface will always be equal to 2, regardless of its level of development. In the usual method of counting boxes it is vital whether some part of the analyzed set is in a box. The result is a single fractal dimension. However, in the multifractal analysis based on boxes, it is important what part of the set is located in the box (detailed explanations are provided in the section on multifractal analysis). The result is a whole spectrum of fractal dimensions [45, 51-53].
The capacitive dimension for these surfaces equal to 2 is a correct result, and the width and shape of the multifractal spectrum further characterize the analyzed surface.
The applied method reflected the correct values of the generalized dimensions [D.sub.q] values for q = 0, 1, 2 (all with average [+ or -] standard deviation) associated for the surface roughness of the CL. These [D.sub.q] values should be taken into account in the engineering design and clinical applications of CL.
Multifractal analysis, when compared to fractal analysis, is a more sensitive and powerful method to quantify the complexity of CL surface roughness, and certifies the presence of areas of different values of the fractal dimension.
The multifractal spectrum allows the assessment of CL regions, both locally and globally, regarding the distribution of the concentration of nanoasperities with different orders of size. It characterizes the geometrical features of the studied surfaces in a more complete and accurate way, describing the amplitude of irregularities on the surface and the degree of their arrangement.
The multifractal spectrum quantitatively characterizes the distribution of the concentration of nanoasperities with different orders of size with asymmetry to the right and left, indicating the domination of small and large values, respectively.
All singularity spectra in Fig. 12, are characterized by a concave down shape, but they present very different patterns regarding symmetry features.
The multifractal singularity spectrum f([alpha]) allows the assessment of the local scaling properties of individual point elevation data sets. The symmetry/asymmetry of the singularity spectrum provides information about the structure-homogeneity/heterogeneity.
The width of the multifractal spectrum (w = [[alpha].sub.max] - [[alpha].sub.min]) indicates overall variability. The wider the singularity spectrum (i.e., the larger the [[alpha].sub.max] - [[alpha].sub.min] value), the higher is the heterogeneity in local scaling indices of the measure and vice versa. The abundance of the measure is reflected in the branch length of the f([alpha]) spectrum. The amplitude of changes around the maximum value of f([alpha]) is a measure of the symmetry of the singularity spectrum. The differences ([[alpha].sub.max] - [[alpha].sub.0] and [[alpha].sub.0] - [[alpha].sub.min]) indicate the deviation of the spectrum from its maximum value (q = 0) toward the right side (q < 0) and the left side (q > 0), respectively. In all the studied cases the left branch of the f([alpha]) spectrum was longer than the right one. A longer left branch suggests that peaks of height readings were less frequent.
Low f([alpha]) values are associated with rare events (extreme values in the distribution), whereas the highest value of f([alpha]) is the capacity dimension, which is obtained by assuming uniform distribution in all the structures.
The multifractal spectrum in Fig. 12a looks like a resultant of the spectra from Fig. 12b-e, which is the expected result and shows that multifractal analysis is very sensitive to local heterogeneity despite a fourfold reduction in the amount of information for the surface area of 1 [micro][m.sup.2] (Fig. 8b-e) relative to the surface of 4 [micro][m.sup.2] (Fig. 8a). The analysis of fractal dimensions from Table 3 leads to similar conclusions.
In summary, the most frequent pattern of the f([alpha]) spectrum was an asymmetrical curve with a greater tendency toward the right side, where q < 0, which indicated the dominance of small nanoasperities. The values of q > 0 are directly associated with the measure that, in our case, is in turn related to the height reading. A correlation between the f([alpha]) spectrum and the mechanical properties of the corresponding samples is then suggested.
The study results show that multifractal dimensions of displacement constitute an effective method with which to evaluate and predict valuable information about biomaterial surfaces and potential biomaterial interactions.
Furthermore, the multifractal spectrum can be used in elastohydrodynamic lubrication to determine the mechanisms of diffusion, attachment, and slow-activated detachment of particles from the CL surface. These predictions could be used in the contact of realistic surfaces with random multiscale roughness (CL and the outer layers of the ocular surface) that reflect the mechanisms of wear.
The surface roughness has an important influence on CL biocompatibility. A detailed understanding of the CL surface geometry is essential in avoiding corneal complications due to interface interactions between the CL biomaterial and the ocular environment.
AFM and multifractal analyses are indispensable tools in the investigation of CL surface quality, and may assist manufacturers in the development and manufacture of CL with high-performance surface characteristics.
In further work, the authors will try to correlate the width of the multifractal spectrum with the surface stereometric parameters.
The following volume parameters (surface): Vmp, Vvc, Vmc, and Wv are calculated: Vmp-volume of material in the peaks, calculated in the zone above cl; Vmc--volume of material in the core or kernel, calculated in the zone between cl and c2; Vvc--volume of void in the core or kernel, calculated in the zone between cl and c2; Vvv--volume of void in the valleys, calculated in the zone below c2.
The authors like to thank Assoc. Prof. Mihai Talu, Ph.D. Eng. from The University of Craiova, Faculty of Mechanical Engineering, Department of Applied Mechanics, Calea Bucuresti St., no. 165, Craiova, 200585, Dolj, Romania, for his constructive comments.
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Stefan Talu, (1) Sebastian Stach (2)
(1) Department of AET, Technical University of Cluj-Napoca, Faculty of Mechanical Engineering, Discipline of Descriptive Geometry and Engineering Graphics, 103-105 B-dul Muncii St., Cluj-Napoca 400641, Cluj, Romania
(2) Department of Biomedical Computer Systems, University of Silesia, Faculty of Computer Science and Materials Science, Institute of Informatics, ul. Bedzinska 39, 41-205 Sosnowiec, Poland
Correspondence to: Stefan Talu; e-mail: email@example.com
Published online in Wiley Online Library (wileyonlinelibrary.com).
TABLE 1. Results of the f([alpha]) spectrum in the range -10 [less than or equal to] q [less than or equal to] 10 with successive 1.0 steps, for scanning a square area of 4 [micro][m.sup.2]. Scanning square area of 2 X 2 [micro][m.sup.2] q [alpha](q, r) f(q, r) -10 2.0983 [+ or -] 0.0234 1.7400 [+ or -] 0.0650 -9 2.0956 [+ or -] 0.0229 1.7659 [+ or -] 0.0601 -8 2.0923 [+ or -] 0.0222 1.7935 [+ or -] 0.0542 -7 2.0885 [+ or -] 0.0213 1.8226 [+ or -] 0.0473 -6 2.0838 [+ or -] 0.0201 1.8529 [+ or -] 0.0395 -5 2.0781 [+ or -] 0.0187 1.8838 [+ or -] 0.0312 -4 2.0713 [+ or -] 0.0168 1.9146 [+ or -] 0.0227 -3 2.0628 [+ or -] 0.0146 1.9441 [+ or -] 0.0145 -2 2.0523 [+ or -] 0.0118 1.9703 [+ or -] 0.0073 -1 2.0385 [+ or -] 0.0086 1.9906 [+ or -] 0.0021 0 2.0179 [+ or -] 0.0042 2.0000 [+ or -] 0.0000 1 1.9760 [+ or -] 0.0060 1.9760 [+ or -] 0.0060 2 1.8737 [+ or -] 0.0362 1.8158 [+ or -] 0.0539 3 1.7341 [+ or -] 0.0779 1.4694 [+ or -] 0.1581 4 1.6542 [+ or -] 0.0925 1.1948 [+ or -] 0.2096 5 1.6162 [+ or -] 0.0919 1.0258 [+ or -] 0.2131 6 1.5947 [+ or -] 0.0891 0.9084 [+ or -] 0.2077 7 1.5813 [+ or -] 0.0865 0.8218 [+ or -] 0.2024 8 1.5724 [+ or -] 0.0845 0.7557 [+ or -] 0.1982 9 1.5663 [+ or -] 0.0830 0.7036 [+ or -] 0.1943 10 1.5618 [+ or -] 0.0819 0.6610 [+ or -] 0.1900 [[alpha].sub.max] - [[alpha].sub.max] = 2.0983 -1.5618 = 0.5365 Statistically significant difference: P < 0.05. TABLE 2. Results of the f([alpha]) spectrum in the range -10 [less than or equal to] q [less than or equal to] 10 with successive 1.0 steps, for four scanned square areas of 1 [micro][m.sup.2] Scanning square area 1 of 1 X 1 [micro][m.sup.2] Qq [alpha](q, r) f(q, r) -10 2.1322 [+ or -] 0.0237 1.6452 [+ or -] 0.0627 -9 2.1284 [+ or -] 0.0232 1.6807 [+ or -] 0.0585 -8 2.1240 [+ or -] 0.0226 1.7182 [+ or -] 0.0533 -7 2.1187 [+ or -] 0.0218 1.7576 [+ or -] 0.0470 -6 2.1124 [+ or -] 0.0208 1.7987 [+ or -] 0.0398 -5 2.1047 [+ or -] 0.0194 1.8407 [+ or -] 0.0317 -4 2.0954 [+ or -] 0.0176 1.8828 [+ or -] 0.0233 -3 2.0838 [+ or -] 0.0154 1.9231 [+ or -] 0.0149 -2 2.0692 [+ or -] 0.0127 1.9592 [+ or -] 0.0076 -1 2.0503 [+ or -] 0.0094 1.9872 [+ or -] 0.0023 0 2.0224 [+ or -] 0.0045 2.0000 [+ or -] 0.0000 1 1.9715 [+ or -] 0.0061 1.9715 [+ or -] 0.0061 2 1.8690 [+ or -] 0.0312 1.8126 [+ or -] 0.0450 3 1.7339 [+ or -] 0.0583 1.4756 [+ or -] 0.1119 4 1.6381 [+ or -] 0.0675 1.1450 [+ or -] 0.1495 5 1.5874 [+ or -] 0.0711 0.9194 [+ or -] 0.1829 6 1.5619 [+ or -] 0.0733 0.7808 [+ or -] 0.2099 7 1.5486 [+ or -] 0.0741 0.6948 [+ or -] 0.2221 8 1.5409 [+ or -] 0.0739 0.6372 [+ or -] 0.2238 9 1.5359 [+ or -] 0.0734 0.5951 [+ or -] 0.2200 10 1.5324 [+ or -] 0.0726 0.5619 [+ or -] 0.2134 [[alpha].sub.max] - [[alpha].sub.min] = 2.1322 -1.5324 = 0.5998 Scanning square area 1 of 1 X 1 [micro][m.sup.2] Qq [alpha](q, r) f(q, r) -10 2.1385 [+ or -] 0.0228 1.6434 [+ or -] 0.0704 -9 2.1348 [+ or -] 0.0223 1.6788 [+ or -] 0.0655 -8 2.1303 [+ or -] 0.0217 1.7168 [+ or -] 0.0593 -7 2.1249 [+ or -] 0.0209 1.7569 [+ or -] 0.0519 -6 2.1184 [+ or -] 0.0199 1.7989 [+ or -] 0.0435 -5 2.1106 [+ or -] 0.0186 1.8417 [+ or -] 0.0343 -4 2.1012 [+ or -] 0.0171 1.8842 [+ or -] 0.0248 -3 2.0896 [+ or -] 0.0154 1.9245 [+ or -] 0.0156 -2 2.0753 [+ or -] 0.0137 1.9600 [+ or -] 0.0077 -1 2.0569 [+ or -] 0.0118 1.9872 [+ or -] 0.0023 0 2.0283 [+ or -] 0.0075 2.0000 [+ or -] 0.0000 1 1.9591 [+ or -] 0.0125 1.9591 [+ or -] 0.0125 2 1.7635 [+ or -] 0.0807 1.6531 [+ or -] 0.1195 3 1.5517 [+ or -] 0.1349 1.1346 [+ or -] 0.2500 4 1.4746 [+ or -] 0.1411 0.8730 [+ or -] 0.2732 5 1.4515 [+ or -] 0.1390 0.7707 [+ or -] 0.2664 6 1.4415 [+ or -] 0.1361 0.7163 [+ or -] 0.2525 7 1.4355 [+ or -] 0.1333 0.6774 [+ or -] 0.2358 8 1.4311 [+ or -] 0.1307 0.6449 [+ or -] 0.2181 9 1.4277 [+ or -] 0.1285 0.6159 [+ or -] 0.2014 10 1.4249 [+ or -] 0.1265 0.5897 [+ or -] 0.1870 [[alpha].sub.max] - [[alpha].sub.min] = 2.1385 -1.4249 = 0.7136 Scanning square area 1 of 1 X 1 [micro][m.sup.2] Qq [alpha](q, r) f(q, r) -10 2.1582 [+ or -] 0.0236 1.6245 [+ or -] 0.0644 -9 2.1546 [+ or -] 0.0232 1.6587 [+ or -] 0.0599 -8 2.1502 [+ or -] 0.0227 1.6958 [+ or -] 0.0541 -7 2.1449 [+ or -] 0.0220 1.7359 [+ or -] 0.0473 -6 2.1383 [+ or -] 0.0211 1.7786 [+ or -] 0.0396 -5 2.1301 [+ or -] 0.0200 1.8231 [+ or -] 0.0314 -4 2.1200 [+ or -] 0.0186 1.8684 [+ or -] 0.0230 -3 2.1074 [+ or -] 0.0168 1.9125 [+ or -] 0.0148 -2 2.0913 [+ or -] 0.0146 1.9524 [+ or -] 0.0076 -1 2.0696 [+ or -] 0.0119 1.9843 [+ or -] 0.0024 0 2.0345 [+ or -] 0.0068 2.0000 [+ or -] 0.0000 1 1.9522 [+ or -] 0.0105 1.9522 [+ or -] 0.0105 2 1.7685 [+ or -] 0.0546 1.6691 [+ or -] 0.0785 3 1.5947 [+ or -] 0.0905 1.2430 [+ or -] 0.1694 4 1.5186 [+ or -] 0.1039 0.9826 [+ or -] 0.2208 5 1.4871 [+ or -] 0.1071 0.8426 [+ or -] 0.2363 6 1.4700 [+ or -] 0.1068 0.7494 [+ or -] 0.2347 7 1.4589 [+ or -] 0.1056 0.6776 [+ or -] 0.2268 8 1.4511 [+ or -] 0.1040 0.6195 [+ or -] 0.2171 9 1.4455 [+ or -] 0.1024 0.5720 [+ or -] 0.2076 10 1.4414 [+ or -] 0.1010 0.5332 [+ or -] 0.1994 [[alpha].sub.max] - [[alpha].sub.min] = 2.1582 -1.4414 = 0.7168 Scanning square area 1 of 1 X 1 [micro][m.sup.2] Qq [alpha](q, r) f(q, r) -10 2.1215 [+ or -] 0.0245 1.6380 [+ or -] 0.0712 -9 2.1176 [+ or -] 0.0240 1.6747 [+ or -] 0.0667 -8 2.1130 [+ or -] 0.0233 1.7140 [+ or -] 0.0607 -7 2.1074 [+ or -] 0.0224 1.7554 [+ or -] 0.0532 -6 2.1008 [+ or -] 0.0211 1.7986 [+ or -] 0.0446 -5 2.0928 [+ or -] 0.0194 1.8426 [+ or -] 0.0352 -4 2.0831 [+ or -] 0.0173 1.8860 [+ or -] 0.0255 -3 2.0713 [+ or -] 0.0146 1.9270 [+ or -] 0.0161 -2 2.0570 [+ or -] 0.0114 1.9627 [+ or -] 0.0080 -1 2.0392 [+ or -] 0.0076 1.9890 [+ or -] 0.0022 0 2.0159 [+ or -] 0.0031 2.0000 [+ or -] 0.0000 1 1.9813 [+ or -] 0.0037 1.9813 [+ or -] 0.0037 2 1.9220 [+ or -] 0.0164 1.8897 [+ or -] 0.0235 3 1.8335 [+ or -] 0.0349 1.6667 [+ or -] 0.0698 4 1.7484 [+ or -] 0.0473 1.3709 [+ or -] 0.1139 5 1.6924 [+ or -] 0.0494 1.1216 [+ or -] 0.1291 6 1.6591 [+ or -] 0.0463 0.9397 [+ or -] 0.1255 7 1.6383 [+ or -] 0.0422 0.8054 [+ or -] 0.1224 8 1.6248 [+ or -] 0.0390 0.7046 [+ or -] 0.1273 9 1.6159 [+ or -] 0.0368 0.6289 [+ or -] 0.1366 10 1.6099 [+ or -] 0.0354 0.5720 [+ or -] 0.1454 [[alpha].sub.max] - [[alpha].sub.min] = 2.1215 -1.6099 = 0.5116 Statistically significant difference: P < 0.05. TABLE 3. The generalized dimensions ([D.sub.q]) for q = 0, 1, 2, all with average [+ or -] standard deviation, for scanning square areas of 1 and 4 [micro][m.sup.2] . No. Scanned area 1 Scanned area of 2 X 2 [micro][m.sup.2] % Scanned area 1 of 1 X 1 [micro][m.sup.2] 3 Scanned area 2 of 1 X 1 [micro][m.sup.2] 4 Scanned area 3 of 1 X 1 [micro][m.sup.2] 5 Scanned area 4 of 1 X 1 [micro][m.sup.2] No. [D.sub.0] [D.sub.1] 1 2.0000 [+ or -] 0.0000 1.9760 [+ or -] 0.0060 % 2.0000 [+ or -] 0.0000 1.9715 [+ or -] 0.0061 3 2.0000 [+ or -] 0.0000 1.9591 [+ or -] 0.0125 4 2.0000 [+ or -] 0.0000 1.9522 [+ or -] 0.0105 5 2.0000 [+ or -] 0.0000 1.9813 [+ or -] 0.0037 No. [D.sub.2] 1 1.8158 [+ or -] 0.0539 % 1.8126 [+ or -] 0.0450 3 1.6531 [+ or -] 0.1195 4 1.6691 [+ or -] 0.0785 5 1.8897 [+ or -] 0.0235 Statistically significant difference: P < 0.05.
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|Author:||Talu, Stefan; Stach, Sebastian|
|Publication:||Polymer Engineering and Science|
|Date:||May 1, 2014|
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