Polymer Latexes - American Chemical Society

Chapter 19

Measuring Particle Size and Size Distribution from Micrograph Images Jason J . Ruan

Downloaded by UNIV OF ARIZONA on August 6, 2012 | http://pubs.acs.org Publication Date: June 3, 1992 | doi: 10.1021/bk-1992-0492.ch019

Reichhold Chemicals, Inc., P.O. Drawer K, Dover, D E 19903

A method is developed for measuring agglomerated and dispersed particle size and size distribution from electron micrographs. The chord lengths of horizontal lines scanned through the micrographs are measured. The chord distribution is smoothed and converted to particle size distribution as if all particles were deagglomerated. A mathematical procedure is derived for converting chord distribution to particle size distribution. Non-linear regression curve fitting is applied using a Gaussian or Lorentzian function. The technique has been verified by computer simulation and implemented on a variety of latex samples. All results show that this technique gives excellent representations of the actual particle size distributions. The whole process is completely automated and takes less than half a minute for each frame of an image. This method is specially designed to resolve agglomerated particles and also applicable to micrographs of fracture surface for the core-shell type structure and/or solid foam analysis.

Particle size distribution (PSD) is one of the most important parameters in characterizing a wide variety of process materials and final products such as emulsions, paints, coatings, adhesives, rubbers, and ceramics. Rheology, porosity, elasticity, solubility, viscosity, stability, permeability, and opacity are some of the variables strongly affected by particle size and size distribution. For instance, two emulsions may have the same composition and average particle diameters and yet exhibit quite dissimilar behavior because of differences in their size distribution. For this reason, many methods have been developed to measure particle size and size distribution^, including the more recent ones such as Sedimentation Field-Flow Fractionation^ and Capillary Hydrodynamic 0097-6156/92/0492-0289$06.00/0 © 1992 American Chemical Society In Polymer Latexes; Daniels, E., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1992.


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Fractionation^). Microscopic techniques probably are still the most reliable and widely used methods. However, microscopic methods have been slow and tedious until the advent of the computer-image analyzer. The major problem of this technique is that it is not able to analyze coagulated or agglomerated particles because they are treated as a single object. Even the very sophisticated systems with the feature of deagglomeration of spheres can only analyze barely touching and very narrow distribution particles. To separate particles from overlapping ones is an extremely complicated and slow process even using robust mathematical operations and normally involves image enhancement, transformation, segmentation, and reconstruction^. The separation often occurs at the wrong place and the treated image does not accurately represent the original particles. An alternative method (patent pending) for particle size analysis was developed based on the correlation between the circle diameter distribution (CDD) and chord length distribution (CLD). Even though the correlation is straightforward, the C L D being an integral transform of the CDD, a mathematical procedure is derived for inverting the CLD to obtain the CDD. The computer automation is performed by a two step process. In the first step, chords are measured by defining two end points of each peak and selected by comparing against the predefined criteria. In the second step, the chord data is assembled in a histogram, smoothed using binomial convolution or non-linear regression, and then converted to the CDD. Since chord lengths are much easier to measure than diameters, and because it lends itself readily to computer automation, the approach presented here as shown in the following examples is an effective technique for the quick determination of particle size and size distribution, especially for resolving agglomerated particles. Mathematical Analysis In particulate micrograph images, the circles are the projection of particles, and hence PSD is the same as CDD. Let F(x) represent the normalized PSD, where x is the diameter. The probability, G(x)dx, that circles in the diameter range (x, x+dx) intersect a random line is proportional to their diameters, hence G(x)dx

xF(x)dx r%Eww

x v

F(x)dx

(1)

where is the mean diameter. For a line which intersects a circle of diameter x, the probability that its distance from the center lies in the range (r,r+dr) is 2dr/x, giving a chord length v=(x -4r ) . Therefore, the probability that circles in the range (x,x+dx) are intersected to produce chords of length in the range (v, v+dv) is 2

2

1/2

In Polymer Latexes; Daniels, E., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1992.

19. RUAN

Measuring Particle Size and Size Distribution

G(x)&-»ft-.

v F ( x ) d x d v

291

(2)

xJx -V y]x -V 2

2

2

1

Any circles so long as their diameters are greater than or equal to chord length, v, might be intersected by a line and produce a chord, v. Summing all the possibilities, we have the CLD,

^ ^ - r - ^ L d x

(3)


< x > J v

which is an integral transform of CDD, and the inverse transform as shown in Appendix I is

(4)

Since the denominator of the integral function in equation 4 becomes zero as v approaches x, a computer program can not be directly applied. Integrating equation 4 by parts to remove the singularity, and approximating the integration as the summation, provided that/fvj is a smooth function, we have

FW-2x22£

71 ~

^

W

v l)-2/(v) Av-l)-^b^] v 2v +

+

(5)

where f(v)=g(v)/v, x and v assume the discrete values 1, 2, ... N, and N is the maximum chord length. Computer Automation Micrograph images can be transferred to a computer with a frame grabber either directly from an electron microscope or from a video camera. The analog signals are then digitized into digital intensities called gray levels at each pixel according to their brightness: 0 for completely dark and 255 for completely bright. Figure 1 shows gray level variance along a horizontal line across an image. Each downward peak is corresponding to the intersection of a particle. Hence, chord lengths can be measured by acquiring the distances between the two end points of each peak. The flow-chart as shown in Figure 2 is an outline for auto chord data collection. The program reads pixel data row by row. After locating the left end point, 10, at which the gray-level, P(I), is just below the peak threshold, PTH, it looks for the minimum point, and continues to find the right end point where P(I) is just above PTH. The derivative analysis can also


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1


- Intensity

V V

Pixel

£

0

•

100

1

•

200

300

1

400

Figure 1. Gray-level variance along a line.

C Start ) Read a row of pixel data

10=1

1

1 I=PMIN(I) 1

Figure 2. Flow chart of auto chord data collection.


r-

500

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be carried out to further define or separate chords from the possible overlapped particles. This process was repeated until the end of a line where I=N. It then reads the next line and continues. After collecting all the chords, it goes to the second program, the flow chart of which is shown in Figure 3. The chord data was assembled into a histogram, smoothed by Fourier analysisfj), binomial convolution^), Gaussian, or Lorentzian curve fitting, and finally converted to the PSD.


Experimental A microscope or video camera and a personal computer with aframegrabber and Reichhold Advanced Particle Size Analysis(RAPSA) menu-driven software are all required in order to implement this technique. RAPSA is available from Reichhold Chemicals, Inc. Depending on the sample form and applications, this technique can be applied using different methodologies as shown in Figure 4. In the case of section micrographs of bulk or solid foam, PSD, P(u), is no longer the same as CDD, but their correlation is the same as that of CDD and CLD. Therefore, PSD can be computed via double inversions with the same transform or single inversion with a different transform(7).

(6)

(7) This paper, however, concentrates on latex suspension samples. The latexes were diluted to roughly 1:2000 with distilled water and cured by bromine, osmium tetroxide, or phosphotungstic acid(PTA) if it was necessary. The sample was spotted on a formvar film supported by a copper grid, and dried at ambient temperature. Micrograph images were transferred directly from a JEOL100 T E M with installation of a Gatan 673 wide angle T V camera to a computer with a MetraByte MV1 frame grabber in it. RAPSA software was used in analyzing particles. Over 50,000 chords from about 10 frames of images for each sample were accumulated before inversion. It takes about 20 seconds to transfer and analyze each frame. Normally, no image enhancement (though available) is required, hence, no image distortion or artifact occurs. The direct measurement of circle diameters was performed using a mouse pointer with RAPSA software. The data was then assembled into a PSD histogram. Results Figure 5 displays a micrograph of polystyrene latex standard along with the corresponding C L D and PSD. The chord data was collected using the algorithm outlined in Figure 2, assembled into CLD, and smoothed by averaging


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1 L=X(J) 1

1 K(L)=K(L)+r ORENTZIANI Downloaded by UNIV OF ARIZONA on August 6, 2012 | http://pubs.acs.org Publication Date: June 3, 1992 | doi: 10.1021/bk-1992-0492.ch019

FITTING 1 ^ ^ ^ ^ ICOMPUTEI F(X)

\ Z H 7 ^ ^

|F-B SMOOTH|

FITTING

ICOMPUTEI FfX)

ICOMPUTti

(STOP) Figure 3. Outline for assembling, smoothing, and converting C L D to PSD.

Figure 4. Schematic methodology flow chart.



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over adjacent values with a binomial distribution spanning 9 channels. It was then inverted to PSD using equation 5, which agrees very well with the PSD by manual measurement(Figure 6). Figure 7a gives a broad base of latexes with severe overlapping, which is extremely difficult, if not impossible, to separate using conventional deagglomeration. With this technique, the PSD is readily determined in reproducible quantities as shown in Figure 7b, which also agrees well with the manual measurement(not shown here). This technique has been implemented on a variety of latexes including multimodal, heavily overlapped, broad base, and negatively stained particles. They all give excellent representations of the actual PSDs. However, since the error from manual measurement could be quite significant, it is difficult to evaluate the true accuracy in reference to the manual method. One way to test the accuracy is to use computer simulation as described below. Computer Simulations To test the accuracy and sensitivity of this technique, circles with a given distribution were simulated at random locations within a video screen. The screen image (treated as a micrograph image) was analyzed with exactly the same procedure used in micrographs. After collecting all the chord data and inversion, the computed PSD was compared against the originals. Broad Gaussian and delta (uniform) distribution were employed in the simulation. Figure 8 shows the given and computed PSD along with the simulated circles. They agree very well. The difference in number or weight average is less than 0.2%. Figure 9 shows computed PSD from simulated uniform circles. Again, it is very close to the originals, the diameter of which was set to 2070 A . Smoothing by Regression Gaussian and Lorentzian distribution are probably the two most important functions in describing the distribution of random observations for most experiments. PSD, in general, can be represented as the weighted sum of N modified Gaussian functions,

(8)

with

(9)



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Figure 5. a) Micrograph of polystyrene latexes; b) CLD; c) PSD.

I 0

1

I

1500

3000

& 4500

I 6000

I

7500

Figure 6. PSD by manual measurement. In Polymer Latexes; Daniels, E., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1992.

I D(A)

RUAN



b


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600

1200

1800

2400

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Figure 9. Computed PSD from simulated uniform circles.


D(A)

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maximizing at x=x . FJx/xJ is so chosen that peak half width decreases with n as c(2n+l)' (see Appendix II) and C L D can be expressed as sum of N explicit functions (see Appendix III), n

m

v

(10)

ri >-E

Polymer Latexes - American Chemical Society

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