We conclude that the present type of cubic zeolitic particles offers the matrix geometry and homogeneity needed for preparing inorganic or geologic standards. For preparing standard multi-element mixtures, the achievement Of Uniform cation distribution would need to be shown, but that should be the case if the ion exchange processes occur with sufficient speed.
ACKNOWLEDGMENT Spectrochemical analysis of the zeolites was done by V. Stewart of the Spectrochemical Section, NBS. Assistance in zeolite irradiation was given by D. Becker of the Activation Analysis Section, NBS.
RECEIVED for review May 18, 1971. Accepted July 23, 1971.
Atomic Absorption Spectrophotometry Applied to the Determination of Breakup Statistics for Fractured Stressed MateriaIs Ben K. Seelyl and Anthony F. Veneruso2 Sandia Laboratories, Albuquerque, N . M . 87115 ATOMICABSORPTION is recognized as a valuable and versatile procedure for quantitative determinations of submicrogram concentrations of many elements. This paper describes the adaption of the technique to the determination of the distribution of initial surface area remaining on particles procured from the residue of fractured metal coated plates (4 in, X 4 in. X 0.06 in.) of stressed ceramic material. The term “initial surface area” specificially refers to that face on the particles which, before fracture, was part of one of the 4 in. X 4 in. surfaces on the stressed plates. The problem resulted from a need to assess quantitatively the fracture characteristics of various types of chemically-strengthened glass and ceramic structures. This application of atomic absorption is unique and provides a method for obtaining the probability density function, f(x) of initial area, x, remaining on fractured particles of stressed materials. The technique greatly simplifies the determination of breakup statistics such as the mean, variance, and probability of obtaining particles containing a greater than specified amount of initial surface area. Computer selected samples resulting from the residue of fractured, chemically-strengthened ceramic and/or glass materials provided particles for chemical analyses. The initial surface area of the metal coated particles ranged from about 5.5 X 10-5 sq. in. to 0.03 sq. in. Interest was focused on the initial surface area remaining on the particle for the following reasons: First, to evaluate the destruction of the surface integrity of the stressed material. Second, stressed glass or ceramic materials of constant thickness have been observed to fracture into particles having a uniformly rectangular cross section. Hence, estimates of the mass and total surface area statistics for the particulate residue can be obtained using this technique. An atomic absorption method was chosen because it offered greater simplicity and efficiency in data collection for the total of 15,000 fractured particles involved in this study of stressed materials. Our paper presents the results for 1124 particles from one of the sample materials tested. Measurement data were automatically recorded in %Ton computer punch cards in a format suitable for input to a digital computer used in data reduction. Chemical Analysis Division 5521. Exploratory Systems Division 1213.
Several metals were considered for deposition on the surface of the stressed ceramic materials. Silver was selected because of the following advantages: immediate and complete removal of the film by 1:1 nitric acid, ease of vapor deposition of a uniform film thickness, good adherence to the substrate, silver can be determined with high analytical sensitivity in the air-acetylene flame (under the working conditions used, 0.04 produced 1 % absorption or 0.0044 absorbance), low noise background permitting the amplifier to be operated at the minimum gain setting, low noise-to-signal ratio, improbable chance of sample contamination through the various manipulative procedures, analytical wavelength in the region of low absorption in an optical air path, and minimum drift of the Ag lamp source during the period of analysis. EXPERIMENTAL
Apparatus. A Techtron AA4 Atomic Absorption Spectrophotometer (Techtron Pty. Ltd., Melbourne, Australia) was used to collect all transmittance data. A Techtron Premix AB51 Burner; designed for air/acetylene and having a 10-cm-long slot, 0.5 mm wide was adjusted in the optical path for maximum response. A silver hollow-cathode lamp (Atomic Spectral Lamps, Melbourne, Australia), operated at 5 mA and at the lowest gain setting on the photomultiplier and amplifier was used. A Hamamatsu R213 Photomultiplier, range of 1850-8000 was used to detect the 3080.7-A analytical wavelength for silver. A Dymec Integrating Digital Voltmeter was modified to automatically average two separate transmittance displays of 10 counts/sec. The electronics were by Sandia. This data logging system (Sandia No. SE160) automatically cycled and recorded the atomic absorption transmittance measurements on an IBM card. A Honeywell 19, 6-in. strip chart instrument recorded peak plateaus. ProFedure. Silver was uniformly deposited at a thickness of 4000A over one surface of the 4 in. X 4 in. x 0.06 in. flat ceramic material. By means of a Talysurf Profilometer it was determined that the silver layer was deposited with a uniformity of better than 5%. The silver-coated ceramic sample was placed in a closed holding fixture with a removable collection tray. This tray was ruled with 16 equal-area squares for sampling purposes, and plate fracture was initiated by means of a small diamond saw penetrating to a depth greater than the compressive stress layer of the strengthened material. Release of the potential energy, stored in the stressed plate,
A,
ANALYTICAL CHEMISTRY, VOL. 43, NO. 13, NOVEMBER 1971
1899
I
4
Initiate Measurement Atomic Absorption Spectrophotometer
Figure 1. Complete area-measurement system @
Read Data
Reading
Pushbutton
Sample Solution
Status
* -
20
40
60
80
100
MEASUREMENT (% Transmittance)
Figure 2. Calibration plot resulted in particulate dispersion. The collection tray, containing the residue of fractured material, was then carefully removed. A computer program (1) randomly selected four squares of the tray from which particles should be chosen. Each particle was placed into individual 15-ml polyethylene beakers. A pipet delivering uniform drop sizes was used to place one drop of 1 :1 nitric acid on the particle to dissolve the silver coating. Distilled and deionized water was added by a “Repipet” having a reservoir of 1 liter, preset to deliver a 3-ml volume of water to each sample container. The samples were aspirated into the burner and, at the maximum transmittance plateau on the strip chart recorder, the digital voltmeter was triggered to average two separate counts of 10 readings for a 1-second period. Data for eight samples were automatically entered on one IBM card (Figure 1.) The possibility of the stressed ceramic material containing silver was investigated by treating 6 grams of the uncoated fractured residue with 1:l nitric acid for 6 hours. This solution was then adjusted to the same dilution as the sample solutions (1 volume acid to 60 volumes of deionized water). Silver was not detected when this solution was aspirated into the atomic absorption flame under the experimental conditions used for data collection.
Keypunch
All operating parameters were maintained constant. (A slight periodic adjustment of the amplifier to maintain 100%T was necessary between readings.) The time required to aspirate a sample and record transmittance was approximately 10 seconds. Two technicians were capable of analyzing 400 to 600 samples per day. Significant readings were recorded for surface areas as low as 5.5 X sq. in. Transmittance values on all particles measured in the normal mode were preceded by a plus sign when printed on the IBM card. When a 5x scale expansion was necessary to record transmittance of 95 or more, the printed transmittance values were preceded by a minus sign. This enabled the computer to recognize the change in the scale expansion parameter. Calibration. A 0.010-inch-thick Mylar calibration sheet superimposed with a mask of 20 calibrated disks was placed into a vacuum deposition chamber together with a piece of the chemically-strengthened ceramic. A uniform deposition of 4000 A of silver was made simultaneously on both the ceramic material and the Mylar calibration sheet. Each calibration disk was then placed into individual beakers and processed exactly as the ceramic fragments described above to provide a set of calibration readings on IBM cards for use in data reduction. The transmittance measurements of the calibration disks were used to generate a set of points defining the transfer function of transmittance measurement to equivalent surface area, x . Since the areas involved cover a range of 4 orders of magnitude, the logarithms of the areas were used in the computer program. Figure 2 displays the calibration plot for the disks. The transmittance measurement of each sample area was examined by the computer to determine its position on the calibration function. Linear interpolation was used to compute the area to which the measurement corresponds. The computer program also shuffled the measurements so that the effect of any subjective tendency to place large particles in beakers first was removed. In this way, the computer generated a sequence of independent sample measurements. EXPERIMENTAL RESULTS
The analytical procedure described above was applied to a sample of Corning Glass Works 0315 type chemicallystrengthened glass. The resulting plot of the density function after 1124 measurements is shown in Figure 3. The estimated mean, variance, and probability of finding particles above a given area are also displayed in the figure. Computation Algorithm. The approximation to the probability density function, f(x), which considers K independent samples from the process is in the form of the truncated series expansion (2). A’ .fK(X)
=
C d i
i= 1
(1)
(1) A. F. Veneruso and R. H. Braasch, “Stochastic Analysis of
(2) R. H. Braasch and R. S. Printis, “Probability Density Function
Fractured Glass,” Sandia Corp., SC-RR-69-506, Albuquerque,
and Expected Value for the Distance Between Two Points in a Rectangle,” ibid., SC-RR-67-2956, 1968.
N. M., 1970. 1900
ANALYTICAL CHEMISTRY, VOL. 43, NO. 13, NOVEMBER 1 9 7 1
where ci is a generated sequence of coefficients and +iis one element from a set of N orthonormal functions. The set of orthonormal functions ( @ i } ( i= 1, 2. . ., N) chosen is
4-
I
+i(x) =
2 ia sin XMAX XMAXX
where
5x6
XMAX
i = 1, 2,
. . ., N
0
and The computational algorithm generates a new sequence of coefficients ( ci}n+l based on the previous sequence { ci and a sample x(n) from the process. The algorithm is
ci(n
+ 1)
=
c,(n)
=
r(n){+i[x(n)I
- ci(n>}
(3)
I\ 200
100
The sequence of weighing factors r(n) is
0
.o
.001 ,002
.om
.OM
.M)5
.006
.007
.ma .w
X(Sq In)
The above algorithm has been shown (3) to converge in the limit with probability 1 to the actual density function. The theoretical development of the computational algorithm is given in References 2 and 3. Because of the necessary length and detail involved, the theoretical development and convergence proof for the algorithm will not be repeated here. The convergence of the coefficients given in Equation 3 is a result of applying the random correction term (4,[x(n)] c , ( n ) } , whose expected value is the gradient of the integral square error with respect to the coefficient ci. The weighing factor r(n) ensures that the random correction term is attenuated as the number of iterations increases; hence, a stable convergence is achieved. A computer program ( I ) , called APROXDN, applies the algorithm to the sequence of measurements of particle area, x(i), and produces an estimate of the probability density function of the random variable in the form
In applying Program APROXDN, the following input variables must be specified: N , the total number of functions [ + i in the orthonormal set; a sequence of calibration readings CBN(1). . .CBN(20) obtained from the laboratory procedure previously described; a set of N coefficients c,(o) as a first estimate of the coefficient vector of the probability density function; and the set of measurements of sample areas. The starting coefficients chosen are the first N Fourier series coefficients of the triangle density function
Figure 3. Algorithm results; Prob (area greater than 0.005) = 0.000209, number of samples = 1124, mean = 0.000766, var = 0.000011 This integral can be evaluated using the summation given in Equation 5 with the result
The estimated variance is
By using Equation 5, the estimated variance can be expressed in terms of the coefficients ci as
1
fA(x> =
2 __ XMAX
0 5 x 5 XMAX
(6)
The estimated probability of obtaining a particle containing surface area x greater than or equal to some specified amount is
By replacing f ~ ( x by ) the summation given in Equation 5 and integrating after interchanging the order of summation and integration this becomes
where XMAX is an upper bound on the sample measurement. Determination of Breakup Statistics. The estimated mean is XMAX
&x)
=
xfdx)dx
(7)
(3) R. L. Kashyap and C. C. Blaydon, “Estimation of Probability Density and Distribution Functicn,” “ I E E E Transactions on Information Theory,” Vol. IT-14, No. 4, July 1968.
Equations 8, 10, and 12, incorporated into the program APROXDN, result in the computation of the expected value, variance, and probability without the necessity of numerical integration or further analytic approximations.
ANALYTICAL CHEMISTRY, VOL. 4 3 , NO. 13, NOVEMBER 1 9 7 1
1901
Table I. Test for Accuracy of Approximating Function APPROXIMATING FUNCTION F(X) = ( 25.14566) SIN( 1 PI X ) ( ,08313) SIN( 2 PI X ) ( 9.50476) SIN( 3 PI X ) ( ,83427) SIN( 4 PI X ) ( 5.07105) SIN( 5 PI X ) ( - ,05524) SIN( 6 PI X ) ( 3.32145) SIN( 7 PI X ) ( ,48247) SIN( 8 PI X ) ( 3.59932) SIN( 9 PI X ) + ( .07821) SIN( 10 PI X ) + ( 3.08570) SIN( 11 PI X ) ( -1.17054) SIN( 12 PI X ) 1.69806) SIN( 13 PI X ) ( 1.80507) SIN( 15 PI X ) ,24738) SIN( 14 PI X ) ( ( ,85843) SIN( 17 PI X ) ( ,55243) SIN( 18 PI X ) ( - .34682) SIN( 16 PI X ) ( 1.92326) SIN( 19 P1.X ) ( ,84588) SIN( 21 PI X ) ,58317) SIN( 20 PI X ) ( ( ( ,04065) SIN( 22 PI X ) ( ,88131) SIN( 23 PI X ) ( -1,77070) SIN( 24 PI X ) - .93567) SIN( 26 PI X ) ( 1.27472) SIN( 27 PI X ) ( .14520) SIN( 25 PI X ) ( .68402) SIN( 28 PI X ) ( 1.56795) SIN( 29 PI X ) ( .18973) SIN( 30 PI X ) ( - ,53858) SIN( 32 PI X ) ( ,25824) SIN( 33 PI X ) ( ,05598) SIN( 31 PI X ) ( .52261) SIN( 36 PI X ) ,48828) SIN( 35 PI X ) ( + ( - ,23882) SIN( 34 PI X ) ( -.11512) SIN( 38 PI X ) ( ,55211) SIN( 39 PI X ) ( ,98459) SIN( 37 PI X ) ( ,35913) SIN( 41 PI X ) ( .00661) SIN( 42 PI X ) ,89854) SIN( 40 PI X ) ( ( - ,90884) SIN( 44 PI X ) ( 1.25355) SIN( 45 PI X ) ( 1.50421) SIN( 43 PI X ) ( ,40053) SIN( 47 PI X ) ( - ,09396) SIN( 48 PI X ) ( ,07605) SIN( 46 PI X ) ( ( ,24942) SIN( 49 PI X ) ( ,17808) SIN( 50 PI X ) ( - ,21829) SIN( 51 PI X ) ,30718) SIN( 53 PI X ) ( - ,18873) SIN( 54 PI X ) ( ,09702) SIN( 52 PI X ) ( ,78438) SIN( 55 PI X ) ( 1,16502) SIN( 56 PI X ) ( ,77252) SIN( 57 PI X ) ( ,05663) SIN( 59 PI X ) ( - ,90621) SIN( 60 PI X ) ( ,62597) SIN( 58 PI X ) ( - ,56008) SIN( 62 PI X ) ( -.34322) SIN( 63 PI X ) ( .31943) SIN( 61 PI X ) ( ,83370) SIN( 65 PI X ) ( - ,14402) SIN( 66 PI X ) ( - ,15196) SIN( 64 PI X ) ( ,03312) SIN( 68 PI X ) ( ,66009) SIN( 69 PI X ) ( - ,20788) SIN( 67 PI X ) ( - ,68787) SIN( 71 PI X ) ( ,66876) SIN( 72 PI X ) ( 1.18945) SIN( 70 PI X ) ( ,13781) SIN( 74 PI X ) ( - .33336) SIN( 75 PI X ) ( ,00302) SIN( 73 PI X ) ( - ,14801) SIN( 78 PI X ) ( ,02499) SIN( 76 PI X ) ( 1.02077) SIN( 77 PI X ) ( - .66187) SIN( 80 PI X ) ( - ,85715) SIN( 81 PI X ) ( ,99348) SIN( 79 PI X ) ( ,24096) SIN( 83 PI X ) ( - ,57131) SIN( 84 PI X ) ( --.14523) SIN( 82 PI X ) ( ,13033) SIN( 86 PI X ) ( 1,70514) SIN( 87 PI X ) ( -.45447) SIN( 85 PI X ) ( - ,69577) SIN( 89 PI X ) ( ,12434) SIN( 90 PI X ) ( - ,95123) SIN( 88 PI X ) ( ,60792) SIN( 91 PI X ) ( 1.61550) SIN( 92 PI X ) ( ,57815) SIN( 93 PI X ) ( ,09707) SIN( 95 PI X ) ( -1.70800) SIN( 96 PI X ) ( .20098) SIN( 94 PI X ) ( ( -.94864) SIN( 99 PI X ) ( ,51652) SIN( 97 PI X ) ( -1.11156) SIN( 98 PI X ) -.59045) SIN(lO1 PI X ) ( -2.19725) SIN(102 PI X ) ( -.96919) SIN(100 PI X ) ( ( .97542) SIN(105 PI X ) ( ,43126) SIN(103 PI X ) ( -1.22136) SIN(104 PI X ) ,67725) SIN(107 PI X ) ( - .74020) SIN(108 PI X ) ( 1.83095) SIN(106 PI X ) ( - ,69770) SIN(111 PI X ) ( ,84270) SIN(ll0 PI X ) ( ,20642) SIN(109 PI X ) ( ( 1.25574) SIN(112 PI X ) ( ,59917) SIN(113 PI X ) ( -.14043) SIN(114 PI X ) ( ,66592) SIN(115 PI X ) ( ,14941) SIN(116 PI X ) ( -.28272) SIN(117 PI X ) .51129) SIN(119 PI X ) ( -.57830) SIN(12O PI X ) ( --.13407) SIN(118 PI X ) ( NUMBER OF SAMPLES TAKEN = 2000 MEAN = ,024793 VAR = ,001196
+ + + + + + + + + + + + + +
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+ + + + + +
+ + + + + + + + +
+ + + + + + + + +
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
PROB AREA GREATER THAN 0.025 = ,499663 INTEGRAL FROM 1 TO XMAX = .99673
APPROXIMATION ERROR
CONCLUSIONS
Because the algorithm presented converges to the actual density function only in the limit, there may be error in the approximation after a finite number of iterations. As a check, two thousand samples of a known uniformly distributed process ( 4 ) over the domain 0.0 to 0.05 were used to test the accuracy of the approximation and the calculation of the mean, variance, and probability of obtaining a sample greater than a given amount. Results are given in Table I. Note that the probability of obtaining a sample between 0 and 0.05 (XMAX) was 0.99673 or a n error of 3 . 2 7 z relative to unity. The probabjlity of obtaining a sample greater than 0.025 was 0.499663 (an error of 0.03z). The computed mean is 0.024793. Compared to the actual value of 0.025, the error is 0.81
The technique described in this paper provides a n expedient means for obtaining quantitative information concerning the breakup statistics of various chemically stressedglass and ceramic materials. The technique is applicable for determining the initial surface area remaining on particles procured from the residue of various materials and structures which are subjected to destruction or disposal.
z.
(4) S. Bell and D. B. Holdridge, “The Random Number Generators for the Sandia 3600 and their Statistical ProDerties.” Sandia Corp., SC-RR-67-65, Albuquerque, N. M., Makh 1967.
1902
ACKNOWLEDGMENT The authors gratefully acknowledge Richard H. Braasch for establishing the basic mathematical approach for this application and William J. Andrzejewski and Raymond M. Merrill who collected the data. Personnel from the Rehabilitation Center, Inc., Albuquerque, N. M., assisted in the procedure of particle selection.
RECEIVED for review March 8, 1971. Accepted July 23, 1971. This work was supported by the United States Atomic Energy Commission.
ANALYTICAL CHEMISTRY, VOL. 43, NO. 13, NOVEMBER 1971
