1400
J. Phys. Chem. 1980, 84, 1400-1413
solution at a mole fraction of Me2SO-d6= 0.5, these lifetimes are almost equal. As the temperature is decreased from ca. 340 K to 250 K, the relation between the lifetimes and the spin relaxation times is changed from case I to III.5 In agreement with the expected behavior for case 111, the 270-MHz ‘H-spectra in the low-temperature range show two separate lines with different line widths (spin-spin relaxation times), whereas selective Tl measurements yield simple exponential relationships with identical spin-lattice relaxation times.
(16) (17) (18) (19) (20) (21) (22)
References and Notes (1) N. Btoembergen, E. M. Purcell, and R. V. Pound, Pbys. Rev., 73, 649 (1948). (2) C. H. Langford in “Ionlc Interactions 11”, S. Petrucci, Ed., Academic Press, New York, 1971, Chapter 6; R. Lachmann, I. Wagner, D. H. Devla, and H. Strehiow, Ber. Bunsenges. Phys. Chem., 82, 492 (1978); D. H. Devia and H. Strehbw, Ber. Bunsenps. Phys. Chem., 83, 627 (1979). (3) H. G. Hertz In H. Falkenhagen, “Theorie der Elektrolyte”, Hirzel, Stuttgart, 1971. (4) H. -H. Filldrier, 0. H. Devia, and H. Strehiow, Ber. Bunsenges. Phys. Chem., 82, 499 (19711). (5) J. Frahm and H. -H. FBldner, Ber. Bunsenges. Phys. Cbem., 84, 173 (1980); J. Frahm, submitted for publicatlon. (6) See, for instance, D. Martin and H. G. Hauthal, “Dimethyisulfoxid”, Akademie-Verlag, Berlin, 1971, and references cited therein. (7) G. J. Satford, P. C. Schaffer, P. S. Le-, 0. F. Doebbler, G. W. Erady, and E. F. X. Lyden, J. Chem. Phys., 50, 2140 (1969). (8) G. Brink and M. Falk, J. Mol. Struct., 5, 27 (1970). (9) J. J. Llndberg and C. Majani, Acta Chem. Scand., 17, 1477 (1963). (10) V. A. Syrbu, M. I. Luplna, and M. I. Shakhparanov, Russ. J. Pbys. Chem. (Engl. Trans/.),51, 474 (1977). (11) K. J. Packer and D. J. Tomlinson, Trans. Faraday Soc., 67, 1302 (1971). (12) T. TokWo, L. Menafra, and H. H. Szmnt, J. C k m . phys., 61, 2275 (1974). (13) J. A. Glasei, J. Am. Chem. Soc., 92, 372 (1970). (14) R. E. London, M. P. Eastman, and N. A. Matwiyoff, J. phys. Chem., 81, 884 (1977). (15) R. Lenk, “BrownIan hrbtbn and Spin Relaxation”,Elsavler, Amsterdam,
(23) (24) (25) (26) (27) (28)
(29) (30) (31) (32) (33) (34) (35) (36)
1977; Adv. Mol. Relaxation Processes, 6, 287 (1975). W. A. Anderson and J. T. Arnold, Phys. Rev., 101, 511 (1956). R. Hausser, Z. Naturforsch. A , 18, 1143 (1963). D. E. Woessner, J. Pbys. Chem., 70, 1217 (1966). A. G. Redfieki In “Advances In Magnetlc Resonance”, Vol. 1, J. S. Waugh, Ed.. Academic Press, New York, 1965, p 28. A. Abragam, ”The Principles of Nuclear Magnetism”, Oxford University Press, London, 1961. J. H. van Vieck, Phys. Rev., 74, 1168 (1948). Torrey (Phys. Rev., 92, 962 (1953)) derlved a formula for spin reiaxatlon rates due to translational diffusion whlch Is based on a nonexponentlal correlation function. However, for a dipolar relaxation mechanism the calculated differences in the TI 2 ( ~ o ) curves are astonishingbysmall even if pure translational diffusion (nonexponentlal correlation function) is compared to pure rotational diffuslon (exponential correlation function). In fact, these devlations occur solely In the “extreme narrowing” (UT < 1) region, whereas the effects on spin relaxatibn times observed in this paper primarily occur in the UT, 2 1 regions. H. A. Resing, Adv. Mol. Relaxation Processes, 1, 109 (1967/68). H. Pfeifer in “NMR: Basic Principles and Progess”,P. Diehi, E. Flu&, and R. Kosfeld, Eds., Voi. 7, Springer, New York, 1972, p 53. F. Noack, ibid., Vol. 3, Springer, New York, 1971, p 83. M. Krlfan, Rev. Sci. Instrum., 46, 863 (1975). J. Frahm, Ph.D. Thesis, Mttingen, 1977. H. Strehlow, Adv. Mol. Relaxation Processes, 12, 29 (1978). The mole fraction in the presence of Be(NO,), is expressed as the total number of moles of Me2SO-de(including that bound to Be2+) divided by the sum of the total number of moles of Me2SO-deand water. S. A. Schichman and R. L. Amey, J. Pbys. Cbem., 75, 98 (1971). T. M. Connor, Trans. Faraday Soc., 60, 1574 (1964). U. Kaatze and R. Pottei, Colloq. Int. CNRS, 248, 111 (1976). L. J. Lynch, K. H. Marsden, and E. P. George, J . Chem. Phys., 51, 5673 (1969). 8. Bilcharska, 2. Florkowski, J. W. Hennel, 0. Held, and F. Noack, Biocbim. Blophys. Acta, 207, 381 (1970). E. Kalman, G. Paiinkas, and P. Kovacs, Ml. Phys., 34, 505 (1977); W. S. Benedict, N. Gallor, and E. K. Plyler, J. Chem. Phys., 24, 1139 (1956). H. Strehlow and J. Frahm, Ber. Bunsenges. Phys. Chem., 79, 57 1975); J. S. Leigh, J. Magn. Reson., 4, 308 (1971); A. C. McLaughiin and J. S. Leigh, ibid., 9, 296 (1973). J. Jen, Adv. Mol. Relaxation Processes, 6, 171 (1974); J. Magn. Reson., 30, 111 (1978).
Analysis of Gouy Interference Pattel‘ns from Binary Free-Dlffuslon Systems When the Dlffuslon Coefficient and Refractlve Index Have C1/* and Csi2Terms, Respectively John 0. Albrlghtt and Donald G. Miller” Lawrence Livermore Laboratoty, University of California, Livermore, California 94550 (Received November 13, 1979) Publication costs assisted by Lawrence Livermore Laboratoty
Gouy fringe patterns are treated by numerical integration and analytical techniques for binary electrolyte systems whose diffusion coefficients D and refractive indices n have concentration dependences including 112 and 312 powers. If one initial solution is pure water, skewing from the ideal case (constant D , linear n) is substantial; correction factors are large; and no averages of experimental quantities will yield D(C). Refractive-index effects were much larger than expected. Analytical results include extension of the Gosting-Fujita theory to yield fringe positions and proof that f(n2I3is a proper extrapolation function to get DA. Numerical integration results show (1) that the Gosting-Onsager theory for ideal systems is better than previously believed at high fringe numbers or low total number of fringes, (2) that masks are more important than previously assumed, and (3) that calculated corrections to experimeqtal quantities can be least squared successfully in terms of variables suggested by the analytical theory. As C/(AC/2) increases (C,moves away from infinite dilution), analytical results approach the numerical values. Proper analyses of dilute electrolyte data require several experiments to obtain the concentration-dependent correction factors. Examples for KCl and CaClz are given.
I. Introduction The Gouy method has been one of the most precise methods of measuring diffusion coefficients. Therefore, it is desirable to extend existing theories to handle the ‘Chemistry Department, Texas Christian University, Fort Worth, Texas 76129. 0022-3654/80/2084-1400$01.00/0
important case of diffusion of a dilute electrolyte solution into pure water. In this case, Gouy fringe patterns are skewed from the “ideal”, owing to square root and threehalves power dependences, respectively, of the diffusion coefficient and the refractive index. Here refractive-index effects are much more significant than Previously recognized. 0 1980 American Chemical Society
The Journal of Physical Chemlstry, Vol. 84, No. 11, 1Q80 1401
Analysis of Gouy Interference Patterns
In this paper, we present methods for the analysis of Gouy interference patterns from free-diffusion experiments for binary systems in which diffusion coefficients depend significantly on the square root and first power of concentration and in which the refractive index depends significantly om the three-halves and second powers of concentration. These methods are based on extensions of analytical theories and on results of numerical integrations. In addition, analytical expressions are obtained for relative fringe positions which apply to nondilute electrolyte systems or to systems in which the diffusion coefficient depends on C and C2 and the refractive index depends on C2 and C3. This paper is an extension of our previous study on the analysis of Hayleigh interferometric patterns for the study of the free-diffusion systems,' where the diffusion coefficient was primarily dependent on the square root of concentrationq2 Theories for the study of systems with constant diffusion coefficients are well established for the Gouy method. An approximate ray-optics theory and a more precise waveoptics theory were presented by Kegeles and Gosting3and Gosting and Morrise4 The most precise equations were given in a detailed analysis by Gosting and Onsager6 (whose text references and equations will be denoted by GO). The effect of Concentration dependences of the diffusion coefficient W and refractive index n in a binary system were developed by Gosting and Fujita6(whose text and equation references will be denoted by GF). They expressed D and n by the following Taylor series expansions:
D = D(C)[l + kl(C - C) + k2(C n = n(C)-I-R(C -- C)[l + al(C -
+ ..I
e) + u ~ ( C-
(1)
+ . .I (2)
where (3)
D = Do[l
(5)
and C is the mean concentration of the initial upper and lower solutions. They inserted eq 1-5 into the differential equation for diffusion (Fick's second law) and solved it by a recursive perturbation method. Results were given as power series expansions in AC, the initial difference of concentration (C, - C,) on the two sides of the diffusion boundary. Terms of the series contain products of the coefficients al, a2,kl, and k2 with expressions involving the error function and its derivatives. They are summarized in section IIA below. Although GF limited their further considerations to polynomial dependences of D and n (i.e., C, C2, C3, ...), their equations can be applied to other types of concentration dependence.,' Fujita' has considered the effect of volume change on mixing for Eree-diffusion experiments. For the diffusion of dilute electrolylm into water, this effect appears to be negligible and will not be considered further. We now apply the Gosting-Fujita expressions (eq 1-5) to the combined square root plus linear dependence of D, which may be written
(6)
Here Dois the limiting diffusion coefficient at infinite dilution. The coefficients p and a may be related to the coefficients kl, k2, etc. in eq 1 by application of eq 3. For this purpose, it is useful to define dimensionless parameters G and H such that
(7) ac 1+p w
H=
+ ac
It is then found that
k , ' = -G+ = H 26
c
Higher order terms k , will be proportional to G/C". Insertion of eq 9 and 10 into eq 1 allows the other GF results to be applied to the square root of the concentration-dependent case. If the mean concentration of an experiment, c, is considerably larger than AC/2, then contributions of terms of the order k3 and higher become negligible. Consequently, as will be shown later, the GF results become reasonable approximations for the polynomial, square root, and combined dependences considered here. On the other hand, if C is close to AC/2 and there is any square root contribution, we will see that the approximationsbecome inaccurate for the Gouy fringe system, just as found before for the Rayleigh system.,' Careful analysis of refractive indexes of dilute electrolytes for systems where accurate data exist8tBreveals a concentration dependence of the form
n = no (4)
+ pC1/2+ aC]
+ pC + uC3I2+ rC2
(12)
As will be shown later, the term uC3/2may contribute significantly to the skewing of fringe positions in Gouy patterns from a dilute electrolyte solution diffusing into water. We can also use the GF expressions for this combined three-halves and squared dependence of n on C. Thus the coefficients al, a2, etc. may be related to coefficients p, u, and 7 by applying eq 5 to eq 12. For this purpose dimensionless parameters E and F may be defined such that
F=
(p
+ 1 . 5 ~ c , '+/ ~27c) TC
(14)
It is found that a1
=
E+F
c
a2 =
-E 6c2
Higher order terms a, will be proportional to E/(?,. Just as for the concentration dependence of the diffusion coefficient, the contribution of refractive-index terms of
1402
The Journal of Physjcal Chemjstry, Vol. 84, No. 11, 1980
order a3 and higher will become negligible if C is considerably larger than AC/2 but will be significant when C is close to AC/2. Thus the GF analytical theory will again give inaccurate results when applied to this poorly convergent 3/2 power case. Gosting and Fujita used their results for the polynomial dependence to obtain an expression for the maximum gradient of refractive index in the free-diffusion boundary. From this, one may calculate the position where a ray of light, obeying Snell’s law, will reach the camera plane after passing through the diffusion cell at the level of the maximum gradient. This position is directly related to the “reduced-height-area ratio”, DA (eq 50 of ref 6), where DA is the measured diffusion coefficient in binary systems with constant D and linear n. Gosting and Fujita then related DAto D(C) with an expression of the order (AC)2which involves the terms kl,k2,al, and a2 of eq 1 and 2. They neglected all higher-order terms, noting that the next higher term would be of the order of (AC)4. These would include k3, kq, a3, and u4 of eq 1 and 2 as well as contain ki and ai (i = 1,2). The GF development did not, however, include expressions for positions of fringe minima and maxima in the Gouy interference pattern. Therefore, in section 11, we derive expressions to order (AC)2for fringe positions as functions of kl,It2, al, and a2. From this, a direct theoretical basis is now established for the GF procedure of calculating DA by extrapolating fringe-position data as a function of f(l)2/3. If the concentration dependence is known, these equations also allow the calculation from C, kl, k2, al, and a2 of Dj/D(C) and of “reduced-fringe deviations”1° nj for those fringes which are typically measured on experimental photographs. These theoretical calculations can then be compared with experimental values and ultimately can be used to estimate correction factors for systems sufficiently far from infinite dilution. For diffusion of an electrolyte solution into pure water, we have both C = AC/2 and a square root term. In this case, as noted in the previous article,l all higher terms in eq 1 and 2 which are calculated from eq 3-6 and 12 contribute to, and may be important to, the final perturbation solution even when C is small. Explicit extension of the perturbation portion of the GF theory to terms of higher order becomes essentially impossible beyond k4 and a4. This may be seen from the complicated development of SchOnert,l3who expresses the concentration dependence of a free-diffusion boundary for systems with terms of order 4 in eq 1. All terms higher than 4 are thus essentially inaccessible. Consequently, both the GF results and our extensions can only serve as a useful guide in the dilutesolution case. Because of these analytical difficulties, we have treated the case where C = AC/2 by numerical integration and will present the results of these calculations in section 111. The accuracy of the Gosting-Onsager theory and the importance of masks is discussed. Numerical integration results lead to methods for analyzing experimental Gouy fringe patterns when C = AC/2. Finally, section IV contains an application to data for the systems KC1-H20 and CaC12-H20. For higher valence types such as 2-2, we estimate that the corrections in dilute solutions can be as much as 1 %, nearly an order of magnitude greater than the precision of measurements.
11. Analytical Extensions We now turn to the derivation of new analytical results valid for all but the dilute-electrolytecase. We will develop equations, based on ray-optics theory, for the change of fringe position in the Gouy pattern which occurs when
Albright and Miller
concentration dependence is introduced. These equations will be _essentialto analysis of experimental systems for which C > AC/2 but Co is close enough to 0 that the analytical theory does not converge to only one term. Specifically we wiII consider the change of fringe position from the ideal case (all ai and kjin eq 1and 2 are O)-to the case where al, a2,kl,and k2 become finite while DLC) and R remain unchanged. This latter corresponds to C being constant, but AC increasing from 0. The concentration dependence will thus be introduced through a Taylor series expansion in powers of AC. We will then assume that although fringe positions predicted from ray optics differ measurably from those that actually occur6 (and are predicted by exact wave optics), the relatively small changes in fringe positions due to the introduction of concentration dependence into ray-optics theory will be close to the changes that would actually occur. Thus eq 49 below expresses the position of a fringe as the sum of two quantities. One is the position predicted by existing wave-optics theory for the ideal case where all ai and ki in eq 1 and 2 are 0 (or AC 0). The other is the change in fringe position due to concentration dependence as predicted from ray optics, but in terms of the wave-optics variable. In our analysis it is convenient to work with a fixed number of fringes, independent of concentration dependence or AC. In experimental situations, the inner celI dimension parallel to the optic axis (cell length a) is fixed, and the number of fringes will change with AC. To avoid this problem, we introduce “reduced” quantities for which An (proportional to the total number of fringes) is divided out. In the experimental situation, this corresponds to increasing the cell length a as AC gets smaller. A. Fringe Positions. Expressions for Gouy fringe positions will now be developed from an extension of the approximate ray-optics theory and will be based on the expressions for the refractive-index distribution given by Gosting and Fujita.‘j They find from their perturbation solution of the diffusion equation that
-
(17)
Here n(C)is the refractive index at the mean concentration of the diffusion experiment, and the functions !Pi are related to the coefficients al, a2, kl,and k2 in eq 1 and 2 through the equations presented below: Bo = *o Ql = kl$l
qz=
h2$2
+ a1($0)2
*o =
1 = - -[2(@)2
4
+
(19)
+ k2$3 + 2a1k1$1+0 + u ~ ( $ O ) ~
where
$1
(18)
*
(20) (21)
+ 2z@’@+ (@’)2 - 21
8(@)3 z(l8 - 4Z2)@’(@)’
+ (12 - 42’)
12(3) T
(22) X
Analysis of Gouy Interference Patterns
The Journal of Physical Chemistry, Vol. 84, No. 11, 1980 1403
r
(
(4 -
-->. 12(3)1/2
-
12(3)ll2 -@((3)1/2~) 7r
1
(24)
Here, @ and a' represent the error function and its derivative so tlhat
The "reduced" variable z is the dimensionless time-independent cell coordinate defined by eq 28. Here x is the z = X/2(D(C)t)l/2
(28)
cell position (x = 0 at the starting boundary position), and t is the elapsed time of a free-diffusion experiment. In a Gouy interferometric apparatus, monochromatic light from a horizontal source slit is focused at a camera position after having passed through a diffusion cell. In the ray-optics t h e ~ r ylight , ~ rays which pass through homogeneous iregions of the diffusion cell (i.e., effectively outside the region of the diffusion boundary) reach the camera plane at an undeviated slit image position.l* Rays passing through a free-diffusion boundary, however, are bent by refractive-index gradients toward regions of greater refractive index. This causes a displacement from the undeviated slit image of the position at which the ray reaches the camera plane. This displacement, Y, is given by eq 29.3 Here (dnldx) is the refractive-index gradient
distribution are sufficient to develop expressions for fringe positions. First, however, it is useful to express the refractive index in a normalized form, A, so that the change of A across the boundary,.Aii, will be independent of AC namely UIC
We will assume here that the actual refractive index can be expressed with coefficients R, al, and a2 of eq 2 with all higher-order terms in the equation set to 0. In this case, eq 2 shows that the difference of refractive index, An, of the upper and lower solutions of a free-diffusion boundary is related to AC by eq 32. If n is given by eq 2 with ai =
0 (i > 2), AA will always be unity independent of the value of a2. This normalization is equivalent to adjusting the inner cell length, a, of the diffusion cell as a function of a2 to keep the total number of fringes constant. By applying eq 31 and 32 to eq 17, where the second term on the right of eq 17 is divided by the normalization factor An of eq 32, and expanding the resulting quotient, we obtain 1 A = \ko + -\kl + - (\k2 - az\ko) + 0 2 Ac 2 ( 3 3 1 (33) If we define 7,the normalized displacement of a ray of light, as 7 = Y/An = Y/[2R(AC/2)(1 + U ~ ( A C / ~ ) (34) ~)]
-[
(4")'
then introducing A into eq 29 by means of eq 31-33 yields
az\ko')
at the position x (or reduced position z ) where the ray passed through the diffusion cell. The coefficient a is the inner cell length, and b is the optical distance from the center of the diffusion cell to the camera position. An expreeision for the phase angle of the ray when it reaches displacement Y at the camera plane is given by eq 30.16 Here is the wavelength of light. For free-dif-
x
modulo 2a
(30)
fusion experiments, two rays of light r and r' passing through the cell at equal values of dn/dz will each reach the same position Y, of the Gouy interference pattern at the camera plane. In general, they will not have the same phase angle or be exactly 180' out of phase, so they will not form a fringe maximum or minimum. Equal vallues of an/& occur in the free-diffusion boundary on1 either side of the position of maximum concentration gradient, (anlaz),,. If the diffusion coefficient were constant and the refractive index linear, so that the refractive index gradient was expressed by a Gaussian distribution, then the positions for equal values of (an/&) would be syimmetrical about x = 0. In terms d ray-optics theory, fringe maxima occur when two appropriate rays reach an associated Yj with 8, = Oj,,, and fringe minima occur when 8, = O j , - 7r, where j and j refer to rays ,giving an extremum. These constraints, along with eq 29 isnd 30, and eq 17 for the refractive index
97
(35)
The primes on the functions @[ signify derivatives with respect to z. Substituting A for n and 7 for Y in eq 30 and applying eq 28, 33, and 35 gives
a2q0- z(\k2/ - az\k
