J . Phys. Chem. 1989, 93, 2169-2175 energy -A.E(n) calculated by formulas 11-Fl to 11-F3 for n is plotted against the population on Be of [Be(H20),JZ+, [Be(OH)(H,O),,]+, and [Be(H,O),,]. In other words, Figure 4 shows how much the complex is stabilized by one more water of hydration when the electron population on Be in the complex is X (a value on the x axis). We can see the relatively smooth and simple curve in Figure 4. It suggests that we can analyze the interaction energy between water and Be with various oxidation stages in terms of electron population or charge on Be rather than the formal hydration number. Once we know the electron population in various species, we can estimate the maximum value of the stabilization energy by the next hydration using the relation in Figure 4. For example, the electron population on Be in [Be20HI3+is 2.36, and then it is expected to be stabilized by about 6.4 eV when a water hydrates the complex. Similarly, [Be(OH)2] and [Be3(OH)3]3+may be stabilized by about 1.4 and 4.8 eV, respectively (see Tables I and 11). In reality, more than one water can hydrate these complex ions in the first shell. To estimate the hydration energy of the ions, we may also be able to use Figure 4. If we apply the "octet" rule mentioned above, we can expect that each Be of the complexes [Be(OH)z] and [Be3(0H)3]3+is coordinated with two waters. Thus, the maximum of the total hydration energy is 2.8 and 28.8 eV, respectively, if the additivity is assumed. But, as repeatedly mentioned, nonadditivity is essential to estimate the hydration energy. To use Figure 4 for estimation of the hydration energy of [Be(OH),](H,O), we have to know
2169
the electron population of Be of the complex. We may assume that the increment of the gross atomic population on Be by one-water hydration to [Be(OH)2] can borrow from that of [BeOH]+(H20)2,since the gross atomic populations of both complexes are nearly equal to each other. Thus, the approximate population on Be in [Be(OH),](H,O) is 3.26, and the total hydration energy of the complex [Be(OH),] is 1.4 + 0.8 eV. By use of a similar procedure, the gross atomic population of ( H 2 0 )[Be3(0H)3]3+is estimated to be 2.87, and those of (HzO)[Be20H]3+and (H20)z[Bez0H]3+are 2.68 and 2.85. The total hydration energy of [Be3(0H)3]3+is 3 X (4.8 3.5) eV, and that of [Be2(0H)l3+is 2 X (6.4 + 5.0 + 3.6) eV. Although these estimated values are very rough, nevertheless they are much better than the estimation based on the additivity. An interesting comparison is between the hydration energies of [Be3(OH)3]3+and three [BeOH]'; the latter is 28.77 (a sum of the -AE(n) in Table I), and the former is 24.9 eV. Since the trimerization energy of [BeOH]' is negative (0.67 eV) with the 3-21G basis set, the hydration of the first shell destabilizes the trimer further. Besides, the entropy factor disfavors the trimerization. This is contrary to the conclusion derived from the electrochemical s t ~ d y . ~
+
Acknowledgment. We thank Dr. Yoshihiro Osamura for stimulating discussions and encouragement on this work. We are also grateful for the use of a HITAC M-680H in the computer center of the Institute for Molecular Science.
Nonlinear Regression Programs for the Analysis of Gouy Fringe Patterns from Isothermal Free-Diffusion Experiments on Three-Component Systems John G. Albright*,+ and Donald G. Miller* University of California, Lawrence Livermore National Laboratory, Livermore, California 94550 (Received: May 24, 1988; In Final Form: September 7 , 1988)
New nonlinear least-squares procedures and equations are presented for the analysis of Gouy interferometric fringe patterns from free-diffusion experiments on three-component systems. Nine nonlinear regression programs written by the authors are described and their use is discussed. The new methods of analysis use most or all fringe positions and do not involve extrapolations. They appear to be more accurate and give better diagnostics of each experiment than previous methods of analysis. Certain of the programs allow direct determination of the total number of fringes, J , to 2~0.03.Others can be used to find corrections for the positions of the undeviated slit image to which Gouy fringe positions are referenced.
I. Introduction For binary systems, the first equations developed for the analysis of Gouy fringe patterns from free-diffusion experiments were presented by Kegeles and Gosting.' Subsequent papers by Gosting and MorrisZand Gosting and Onsager3 gave improvements of this analysis. The cases of concentration-dependent diffusion coefficients D and refractive index n in terms of polynomials in C were considered by Gosting and Fujita4 and of polynomials in C1/2by Albright and Miller,5 respectively, for binary systems. Theories for analysis of Gouy patterns from ternary systems evolved in papers by Akeley and Gosting6 and by Baldwin, Dunlop, and G o ~ t i n g . ~General equations were set forth in two papers by Fujita and G o ~ t i n g . ~We , ~ will assume the reader is familiar with the equations in those papers. Unless otherwise specified, notations used here will be those used and defined in the above references. So far there have been no theories for concentration dependence of diffusion coefficients, Dik, and n in ternary systems. 'Permanent address: Chemistry Department, Texas Christian University, Forth Worth, TX 76129. *Address correspondence to either author.
0022-3654/89/2093-2169$01 S O / O
Fujita and Gosting9 gave a completely general method for calculation of ternary Dikrbased on the quantities DA(obtaized from C,) and Qo(obtained from the area under the fringe deviation graph). Subsequently, Dunn and Hatfieldlo and Revzin" developed alternative least-squares procedures for the analysis of Gouy patterns from ternary systems, also based on the equations of Fujita and Gosting. In all of these least-squares methods, the value of C, for each pattern is first obtained by extrapolation of Yj/exp(-z?) v e r ~ u s f ( z ~ tof(zj) ) ~ / ~ = 0, either graphically or by Kegeles, G.; Gosting, L. J. J. Am. Chem. SOC.1947, 69, 2516. Gosting, L. J.; Morris, M. S. J. Am. Chem. SOC.1949, 71, 1998. Gosting, L. J.; Onsager, L. J . Am. Chem. SOC.1952, 74, 6066. Gosting, L. J.; Fujita, H. J . Am. Chem. SOC.1957, 79, 1359. Albright, J. G.; Miller, D. G. J. Phys. Chem. 1980, 84, 1400. Akeley, D. F.; Gosting, L. J. J . Am. Chem. SOC.1953, 75, 5685. (7) Baldwin, R. L.; Dunlop, P. J.; Gosting, L. J. J . Am. Chem. SOC.1955, 77, 5235. (8) Fujita, H.; Gosting, L . J. J. Am. Chem. SOC.1956, 78, 1099. (9) Fujita, H.; Gosting, L. J. J . Phys. Chem. 1960, 64, 1256. (10) Dunn, R. L.; Hatfield, J. D. J. Phys. Chem. 1965,69, 4361. (11) Revzin, A. J . Phys. Chem. 1972, 76, 3419.
0 1989 American Chemical Society
2170 The Journal of Physical Chemistry, Vol. 93, No. 5, 1989
linear or quadratic regression. Graphical methods are of course somewhat subjective. While reviewing data being obtained at LLNL, it became apparent that these methods of extrapolation could introduce errors if Qo(denoted by Q in ref 9) were large or the total number of fringes J were small. Errors arise, in part, because the choice of how many fringe positions to use before switching from linear to quadratic regression is both subjective and uncertain. In addition, when J is small, there are too few fringes for good statistics; in this case the curvature of Y,/exp(-z:) v e r s ~ s f ( z ~becomes ) ~ / ~ so large with increasingf(zj) that even a series with a quadratic term cannot properly fit the data. The above reasons provided one motivation for developing a new set of nonlinear least-squares programs, in which the complete theoretical equation for Gouy fringe positions as a function of fringe number is fit to each Gouy fringe pattern. The quantity C,is included as one of the adjustable parameters that are simultaneously varied to make the equation fit most or all the fringe position data of each pattern. This approach gives better results and better diagnostics for the analysis of Gouy patterns than previous methods.e11 A second motivation for developing these new nonlinear least-squares procedures was based on problems with a new method for obtaining J for each pattern by extrapolating Y, data of the high numbered fringes.I2 This method, denoted by PQ, works well and is theoretically exact for binary systems with constant D or for ternary systems where Qois zero. However, the method is only approximate for Qo not equal to zero and has the same kind of extrapolation problems with high numbered fringes as C, does with low number fringes. These problems can again be avoided by using the complete theoretical equation in which J is now an additional parameter. Again, all the measurable fringe positions of a given pattern can be used to find J, C, and the other parameters simultaneously. The analysis described here is for each pattern of each experiment in a set of experiments, all at the same mean composition. Various quantities, defined below, are obtained by least-squares fits to each pattern and include C,, u- (or u+), r- (or r+),and J. The quantity C, is associated with each pattern. The u-, r-, and J should be common to each experiment. Therefore, the agreement of each of these three from pattern to pattern is a measure of the precision of that experiment. Moreover, u.. (and u+) are common to all the experiments of a set. Therefore, the agreement of the average value of CL (and u+) from experiment to experiment is a measure of the precision of the set. These concepts have led to the development of nine different programs for various conditions. Their description and use are presented below. 11. Theory Fujita-Gosting Equations. We assume ideal ternary diffusion applies so that the equations of Fujita and Costing are ~ a 1 i d . l ~ Then the basic equations relating fringe positions, 5, of a given fringe pattern to parameters for the description of temary systems are given by
where
d'(P) = ( 2 / 9 W exp(-P2)
(2)
Here, the quantities y j are implicitly defined by the equation
+
r+f(a+1/2yj) r-f(a-1/2yj)= f ( z j ) =
(zj/n
Albright and Miller of fringes J (see ref 2, 3, and 14), and where the functionf(B) is defined by the equation
f(P)
(4)
As noted in ref 9, u+, u-, and the set of I"+ values from the set of experiments, along with the refractive-index increments (an/aCi) = Ri, provide sufficient constants to calculate the four Dit. The reduced-height-area ratio, DA, is related to r+,r-, u+, and u- by the equation 1/DA1l2= Moreover,
r+u+1/2 + r-u-1/2
r+and r- satisfy the equation8 r++ r- = 1
(5)
(6)
Number of Independent Variables. The six variables of eq 1 are I'+, I?-, u+, u-, C, and y j . However, they are Pelated by eq 3,5, and 6. Consequently, the minimum number of independent regression variables is three. Explicit elimination of r+and u+ is easy using eq 5 and 6. However, y j cannot be explicitly eliminated. On the other hand, the nonlinear least-squares equations based on eq 1 have the derivatives of y j with respect to the least-squares variables appearing linearly. These derivatives can be explicitly eliminated from eq 1 by the linear expressions for the y j derivatives obtained from eq 3. Consequently, the three remaining independent least-squares variables are r-,u-, and C, (or alternatively, r+,u+, and C,). It is more convenient to use auxiliary parameters as described below. Another variant is given in Appendix B. Transformation to New Independent Variables. Consider the following auxiliary variable scheme. Multiply both the numerator and denominator of eq 1 by D A 1 / 2and define the new variables S-, S+,and lj by the equations
S+ = ( U + D ~ ) I / ~
(7)
s- = ( u B A ) 1 / 2
(8)
tj = y j / D A ' / 2
(9)
Then eq 1 and 3 may be written
yj = c,(r+s+ exp(-S+21?)
+ rs-exp(-S+2t2))
r+f(s+tj) + r-f(s-tj)= f ( z j ) = ( Z j / J ,
(10) (11)
Equation 5 shows that S+ and S-may be related by the equation
r+s++ r s - = 1
(12)
Equations 6 and 12 show that either I?+ or I?- and either S+ or S-can be eliminated from eq 10 and 1 1 . Consequently, the fringe positions can be related to a different set of three inde,+'I and S+),as pendent variables, e.g., (C,, I?-, and S-)or (C,, well as to the quantity Z j for the fringe minimum j . Comments. The value of Z j depends on the approximation to the wave theory of Gouy fringes. If the Z j of eq 1 1 are given by equations of Costing and Morris,2 they will only depend on the fringe number j . This applies when J > 30, and when the approximation of a Gaussian refractive-index-gradient distribution can be used. Otherwise, given approximate values of three parameters, e.g., r-, s-,and J , one may obtain a refinement of Z j based on the extension of eq A5 of ref 9 and eq 40 and 41 in Gosting and O n ~ a g e r . ~ J ~ (14) We use the notation I, in place of the usual (IS) Equation A5 of ref 9 IS written j
(3)
= erf (PI - ( 2 8 / ~ ' / ~exp(-PZ) )
S,.
+ y4 - (1/r'I2)(W2/v2) + ( ~ / K ' / ~ ) ( W ~ /=Vf l~z )j )~= Z , / J
where 2,is a function of the fringe number j and total number
The W2and V,, given by eq 40 and 41 of ref 3, are functions of the derivatives with respect to tj of function Ip, which for a ternary system can be written
(12) Miller, D. G.;Sartorio, R.; Paduano, L. Manuscript in preparation. ( 1 3) Ideal diffusion is the case where the diffusion coefficients are constants independent of concentrationand the refractive index is a linear function of the two solute concentrations.
Once values of r+,l,' S+,and S_are fust obtained in the nonlinear regressions using Z,approximations from ref 2, eq A5 of ref 9 may then be used to find revised values of Z,. This in turn leads to revised r+.etc. If needed, the process may be iterated.
= w + m + t+ ) rm-m
Analysis of Gouy Fringe Patterns
The Journal of Physical Chemistry, Vol. 93,No. 5, 1989 2171
TABLE I
program
independent variables
F3 F2P F2M F4 J F3JP F3 J M F4Y F3YP F3YM
c,, r-,sc,, r+ c,, rc,, r-,s-, J c,, r+,J c,, r-,J c,, r-,s-, Y, c,, r+,Y, c,, r-, Y,
conditions when used always used first P+ small
r- small
J uncertain J uncertain and J uncertain and
r+small r- small Y,uncertain Y, uncertain and r+small Y, uncertain and I'- small
For experimental cases for which we have used the new nonlinear least-squares procedures, the Gostingansager refinements in Zj have been negligibly small, even for the lowest fringe numbers 0' = 0 or 1) where the values of Z, are most dependent on the values of J . However, if Qo were very large or J small, such refinements might be needed in the last iterations of the leastsquares procedure.I6 If J is also a desired quantity, it represents a fourth independent parameter. It appears explicitly in eq 11 and implicitly determines the values of f j . There are occasionally experimental problems in determining the undeviated-slit-image position for y/ as described in section 1II.D. A correction, Y,,can be treated as another fourth independent parameter. However, because Y, and J are strongly correlated, only one can be chosen as the fourth parameter. The authors note that an alternative least-squares procedure is possible, analogous to the above, which directly uses the more familiar parameters given in ref 9. In this case, the independent variables become C,, r+,and p , where p is defined by p = (S-/S+) = ( u - / u + ) l / 2
(13)
Also, J and Y,can be independent variables in a manner similar to the above developments. The equations needed for this least-squares procedure are given in Appendix B. We have not yet written programs based on these variables and thus cannot comment on convergence properties at this time. 111. Nonlinear Least-Squares Programs A. General Description. Equations 6 and 10-12 form the basis
for nonlinear regressions that fit the fringe positions from a Gouy fringe photograph to some minimum number of parameters. We have written nine variations of such nonlinear regression programs, which are summarized in Table I. The necessary derivatives for these nonlinear least-squares programs are listed in Appendix A. 1. F3 Program. The first program, F3, fits fringe position data to C,, I'-, and S-. This is appropriate when J is known, which is usually the case; when lQolis larger than 25 X lo4; and when r- or r+is not close to zero. We sometimes have problems using this program when r- or r+is close to zero, although it usually works even then. DA for each pattern can be calculated from C, and the corrected time (see below), and Qocan be calculated from C,, r-, and S-.9 2. F2 Programs. If lQol is less than 25 X lo4, then the three constants of F3 are less well determined. Reducing the number of constants can improve this situation. Therefore, the authors have also written two programs, F2P and F2M, that fit fringe position data to (C, and r+)and (C, and r-),respectively. In these cases, the sets of DA and Qo for all experiments are obtained from initial calculations using the three-parameter program F3. By means of the standard Fujita-Gosting procedure: these DA and Qoare used to calculate the Dik,u+, and u- for the whole set of experiments, as well as calculate values of DA, r+, and r- for each experiment. For those experiments where either r+or r- is close to zero, either CJ+or u-,respectively, and DA can now be used in eq 7 or 8 to fix the value of S+ or S- to be used in one of the two-parameter
programs. More specifically when r+is close to zero, a fmed value of S+ is used in program F2P to fit C, and r+to the fringe data. In this case, the fit is not sensitive to the value of S+,so its value need not be known too precisely. For similar reasons a fixed value for of S- may be used in program F2M to obtain C, and experiments for which r- is small. 3. F4J Program. The total fringe number, J , is normally obtained in two parts: from initial fractional part of a fringe photographs to give the fractional part of J , and from either counting fringes in Rayleigh photographs or using the PQ programI2 to give the integer part of J . However, if there are experimental problems, J can also be calculated as an additional parameter in the nonlinear regression to the Gouy fringe patterns. Program F4J fits fringe positions to J , C,, r-,and S-. This has proved useful in determining the value of J to f0.03 or better if lQolis large, if r- is not close to 1.0 or 0.0, and if the general fit of fringe patterns is better than f 3 Fm. When this program is used, the values of J are averaged from all the fringe patterns for an experiment. However, the other three parameters are less well determined in a four-parameter fit. Therefore, this average value of J is now taken as fixed, and better values of the other parameters are redetermined from the three-parameter regression F3. 4. F3J Programs. The problems noted above for program F3 when lQolis small and r+is near 0 or 1 are more exaggerated for F4J. Reducing the number of parameters again improves the situation. Therefore, two programs, F3JP and F3JM, have been and ( J , C,, and I?-), respectively, written to fit ( J , C,, and r+), to fringe position data. The same criteria used above in the fixed-J cases will determine when to use r+or r-. These programs have been used to find J when r+is nearly zero or close to unity. Again, values of J from all the fringe patterns are averaged, and this average value is then used in the two-parameter regressions F2P or F2M, in which J is not a parameter. 5. F4Y and F3Y Programs. It is possible to include an adjustment to the undeviated-slit-image position. Such an adjustment may be required owing to either a problem in the initial 6 correction17or to problems in photographing the two reference fringe positions associated with each Gouy fringe pattern,I7 as described in section D. This adjustment, denoted by Y,,can be treated as an additional adjustable parameter in the nonlinear least-squares procedure. We have written three such programs: F4Y with (Y,, C,, r-, and S-); F3YP with parameters (Y,, C,, r+);and F3YM with C,, and r-).The choice of program to use follows parameters (Y,, the same criteria as described for cases without Y,. In using our programs, we found that a least-squares program where both Y,and J are simultaneously adjustable was not useful because Y,and J are highly correlated. B. Structure of Nonlinear Regression Programs. Step 1. At. The correction, At, to the clock time due to the inevitable imperfect starting boundary is obtained as follows. The quantity C,* is first obtained by averaging Ctj for fringes 1-15 (omitting fringe 0 because its position is less certain), where each Ctj is calculated from Ctj = Y,/exp(-zj2)
(14)
and where z, is in turn obtained from the rightmost equality of eq 11 by iteration of eq 4. The quantity D,* is calculated from this average for each fringe pattern, treating the system as if it were an ideal binary,13 using eq 16 below. Then a straight line is fitted to D,* versus l / t r by linear regression, where t r is the experimental clock time. The slope of the line is D*At and the intercept is D* (see ref 1 and 18), thereby yielding the corrected time t = t ' + At
(15)
Z,in the early cycles of the nonlinear least-squares regression sometimes leads
(17) Gosting, L. J. J . Am. Chem. SOC.1950, 72, 4418. (18) This would be the normal way to calculate the apparent diffusion coefficient for a binary system, but it will not give the correct value for the apparent reduced-height-area ratio D,' for a ternary system; therefore it is
to divergence.
marked with an asterisk.
(16) We have found that including the Gosting-Onsager refinement for
2172 The Journal of Physical Chemistry, Vol. 93, No. 5, 1989
Step 2. First Approximation to C,for Each Pattern. An initial value of C,for each pattern is obtained by a linear extrapolation ) ~ / ~ = 0. The outer eight fringes of C, (eq 14) v e r ~ u s f ( z ~ tof(z,) (fringes 0-7) are satisfactory for this purpose. Step 3. Regression of C,, r-, and S- for Pattern I . For the regression to obtain C,, I?-, and S-, values of r- and S- of 0.99 each and 1.01 each along with the step 2 value of C,are tried. Whichever gives the best fit is used as a starting point of the nonlinear regression procedure sketched in Appendix A. Keeping C, fixed at the step 2 value, the regression proceeds until r- and S- are adjusted to give a minimum of the function F of eq A4 in Appendix A. At each iteration of this regression, after new approximations of values of I?+, r-,S+,and S- have been obtained, new values of (,are determined from eq 11 by the NewtonRaphson method. To prevent the regression routine from becoming divergent, I’- and S- are constrained to be both less than unity or both greater than unity. Step 4 . Unconstrained Regression. Now all of C,, r-, and S(F3 program) [and J (F4J program) or Y, (F4Y program) if included] are allowed to vary until a minimum of F (eq A4) is reached. At this stage, fringe positions that are more than 20’ 4)/G 2) standard deviations outside the calculated value are discarded, and the regression for this pattern is then completed without these points. Step 5 . Subsequent Patterns. After values of r- and S- are found for the first pattern, they and the C, (from linearly extrapolating the first eight fringes) for each subsequent pattern are used as initial values for regression of this subsequent pattern. The regression then proceeds as in step 4. Once the regressions for all patterns are complete, DA is calculated for each pattern from apparatus constants, and from J , C,, and the corrected time t , using the equation’
+
+
DA = ( J X b ) z / ( 4 ~ C ~ 2 t )
where h is the wavelength of the light and b the optical distance from the center of the cell to the photographic plane. The values of I?-, S-, and DA (the latter calculated from eq 16) in principle should be the same for each pattern. The deviations of these quantities from pattern to pattern help indicate the internal consistency and precision of the experiment. When Qois small or when r+(or r-)is small, (less than 0.1) then the programs that fix S+ (or S-) are used, e.g., (F2P, F2M), (F3JP, F3JM), or (F3YP, F3YM). In these cases, I’+ (or I?-) is the only variable adjusted in the third step of the regression. For the first pattern, r+(or I?-) starts at unity but is estimated from previous results for later patterns. In the final stage of the regression, C, and I’+ (or F-), plus J or Y, if included, are adjusted in programs F2P, F2M; F3JP, F3JM; and F3YP, F3YM, respectively. C . Output from the Nonlinear Regressions. I . I;. Residuals. As a diagnostic, the difference between measured and calculated I;. is printed for each fringe position and is used to calculate the standard deviation of the fit to a given pattern. 2. DA. The values of DA calculated for each pattern by eq 16 are printed. These values can be averaged as noted above,lg and the standard deviation of these values of DA gives another diagnostic for experimental precision. Values of C,from each pattern are also used to calculate the apparent reduced-height-area ratio, DA’. By conventional procedures,2 these DA’ are extrapolated versus the reciprocal of the elapsed time, l/t’, to l / t ’ = 0 to yield the extrapolated value of DA for the experiment. This extrapolated D A can be compared with the average DA. The slope is DAAt, and this At can be compared to the At obtained from step 1 of section B. 3. Qo. Once I?+ and S+ (or r- and S-) are determined for a fringe pattern, these values are used to calculate the area under (19) This average value of DA could be used in the final calculation of the four D,k, but we preferred to use values of DA obtained by direct extrapolation of DA‘, the apparent values of DA of section C.2, because trials show them to be more consistent within a set of experiments. For good experiments, the difference between the two values of DA is usually small.
Albright and Miller the Q graph, Qo, by using eq 4 and 1 3 of Fujita and G o ~ t i n g . ~ . ~ ~ The values of Qo from each pattern are then averaged to obtain the value of Qo for the experiment. Values of Qo obtained in this way appear to be better than obtained by earlier methods. In the earlier methods, an average fringe deviation for each numbered fringe is obtained, usually with outliers excluded by some criterion. The area under the fringe deviation graph is then found by numerical integration using a method for unequal increments, such as the trapezoidal or Lagrange interpolation rules. In the new methods described here, intercomparison of Qofrom each pattern gives a useful diagnostic for the experiment. For experiments performed at LLNL with the “Gosting” diffusiometer, the standard deviation of the values of Qo obtained from the fringe patterns was usually less than 1 X lo4. Standard deviations greater than 2 X lo4 usually meant there was a problem with the experiment. D. Diagnostics. Analysis of a large number of experiments shows that the diagnostic outputs can indicate a bad experiment, a bad pattern, a bad 6 correction,” a bad 6’ or 8” measurement’? (Le., a bad J ) , or a bad “base line”. Base line denotes the average position of the reference patterns photographed before and after each Gouy pattern.’? These reference patterns determine the undeviated-slit-image position. A “bad” base line is indicated by a significantly different value from those of the other patterns, which results when the two reference photographs are at significantly different positions. This can happen from mechanical jarring or inadvertent movement of the plate or its holder between the two reference photographs. The useful diagnostics are (1) the individual Qo values for all the patterns; (2) the q residuals for each pattern; (3) the standard deviations of q for each pattern and the overall standard deviation of 5,uy, for the set of all patterns; (4) the standard deviation of Qo, uQ;(5) agreement among base-line values; (6) the standard deviation of J , uJ,with J codes; and (7) the standard deviation of Y,, uyc,with Ycodes. All diagnostic examinations begin with F3 programs. If the base lines are small and approximately the same, then the following observations are useful: These are based on the high-precision Gosting diffusiomecer. 1. A good experiment has uy 5 2-3 pm and uQ I (1 .O-1.5) x If u y 2 4-5 pm or uQ 1 2 X there may be justification for repeating it. 2. If there is a trend in the Qo values with time for an experiment, there may be an error in the 6 correction or a leak in the cell assembly: a. If the 6 correction for this experiment is discrepant with those of the other experiments at the same mean composition, then 6 can be taken as the average of the others. If this improves the results, then this value of 6 can be retained. Alternatively, a Y program can be applied. If Y, is relatively constant for each pattern and uycis distinctly smaller than Y,, then Y, can be added to the original 6 to get a revised 6. Seriously discrepant values of Qoor of 5 standard deviations for one or two patterns usually justifies rejection of these patterns in the final analysis. b. If there is a large scatter in 5,especially within early time patterns, then there could be a problem in the formation of the initial boundary or with leaking from the siphoning needle as it is withdrawn. However, if the problem is a distorted boundary, it will “heal” itself with time. Therefore such an experiment can be salvaged by retaining only the later patterns. If the siphoning needle leaks on withdrawal, the experiment is not salvageable. 3. If the 5 residuals all have the same sign for high fringe numbers, then there may be an error in the 6 correction, in J , or both: a. If 6 is similar to those for the other experiments of the set and if these residuals are more or less the same for all the patterns, an error in J is suspected. In this case an appropriate J code (F4J, (20) Following Revzin (see ref 22). we use the notation Qoinstead of the Q used by Fujita and Gosting in ref 9.
The Journal of Physical Chemistry, Vol. 93, No. 5, 1989 2173
Analysis of Gouy Fringe Patterns
TABLE 11: Comparison of Calculations of Di, for Raffinose-KC1-H20Q diffusion coefficients X 105, cm2 s-I D, I f DI9 f D,I f linear 8 0.4314 0.0004 0.0007 0.0002 0.2260 0.0096 all F3 all F2 F3, F2 all F3 all F2 F3, F2
){ ](
DA(av) Qo(av)
DA(eXt) Qo(av)
Units: u+, u-,
0.4324 0.4322 0.4323 0.4322 0.4321 0.4322
0.0002 0.0002 0.0002 0.0002 0.0002 0.0002
s; DA, cm2 s-];
0.0009 0.0009 0.0009 0.0009 0.0009 0.0009
k,cm-’ SI/^.
0.0001 0.0001 0.0001 O.OOO1 0.0001 0.0001 u
0.2190 0.2225 0.2198 0.2195 0.2228 0.2200
0.0037 0.0042 0.0041 0.0036 0.0042 0.0040
D,, 1.8230 1.8211 1.8215 1.8213 1.8197 1.8210 1.8206
f 0.0046 0.0020 0.0021 0.0022 0.0020 0.0020 0.0021
U+
u-
231 864 231 350 231472 231 397 231 431 231 504 231439
54851 54907 54895 54905 54950 54910 54924
u(DA-1/2)b ulk)b 0.2240 0.1556 0.1298 0.1565 0.1649 0.1050 0.1498
0.0942 0.0361 0.0415 0.0401 0.0357 0.0414 0.0395
denotes standard errors of the indicated quantities.
F3JP, or F3JM) is tried. If the difference between the experimental J and average calculated J is more than 2uJ, then the J-code value is a reasonable choice. b. If the I;. residuals for high numbered fringes are of the same sign, but the systematic deviations become larger at later times or 6 is discrepant compared to the other experiments, then an error in 6 is indicated. In this case an appropriate Ycode (F4Y, F3YP, or F3YM) is tried. If the Y, values are relatively constant and their average is somewhat larger than uyc,then Y, can be added to the original 6. Alternatively, an average 6 from the other experiments can be used. Unfortunately, the analysis of trends in I;. residuals is somewhat subjective because Y, and J are highly correlated variables in the least-squares analysis. 4. If one or more of the base-line values are widely discrepant, a Y code can eliminate this problem. 5 . Finally, if the 6 correction or the 6’, 6” measurements are lost, for example by the accidental breaking of a plate, the experiment can be usually salvaged by the use of Y codes or J codes, respectively. E. Determination of the Dlk. The four diffusion coefficients are calculated from values of J , ACl, AC2, DA, and Qo for a set of experiments at the same mean concentrations Cl and C2. We use a regression program originally written by Revzin (private communication2’). It follows procedures outlined by Fujita and G ~ s t i n g and , ~ is described in Revzin’s thesis.22 Our extended version, denoted by RFG, calculates values of u+, u-, the Ri, and the four diffusion coefficients for the set of experiments. The program also gives recalculated values of J , DA, and Qo for each experiment, as well as ai and ri. The RFG program is first used, with its necessary input values for each experiment obtained from the three-parameter F3 program for that experiment. If all Qo are large (IQol > 25 X and if neither of the r+(or I?-) are small, then this calculation yields the final values of the diffusion coefficients. If there are experiments where lQol is small (Le,,
