SILASE. GUSTAFSSON, JULIUS G. BECSEY,AKD JAMES A. BIERLEIN
1016
Thermal Diffusion Measurements by Wave-Front-Shearing Interferometry
by Silas E. Gustafsson,’ Julius G. Becsey, and James A. Bierlein Aerospace Research Laboratories, Ofice of Aerospace Research, Wright-Patterson Air Force Base, Ohio (Received October 28, 1964)
The application of wave-front-shearing interferometry to the study of thermal diffusion is analyzed mathematically, and it is shown that an optimum shear exists which maximizes the sensitivity of the method. Experimental measurements on electrolyte solutions, using a form of interferometer described by Bryngdahl, give results which are coniparable in precision to the best previous determinations made by other techniques. The present niethod has worthwhile advantages of simplicity and convenience.
Introduction The thermal diffusion of binary liquid systems is usually studied as an unsteady-state process, in which the experimenter aims to create a convection-free onedimensional environment of vertical heat and mass transfer in a suitably designed saiiiple cell. The progress of the diffusion can be nieasured in situ by a variety of optical methods-image displacement , 2 - 5 Gouy diff r a ~ t o i i i e t r y ,and ~ ~ ~ Rayleigh interferometry*-all of which derive ultimately from the distortion of a transmitted wave front by the gradients of refractive index attending the diffusion process. High sensitivity is a goal of primary iniportarice in the development of measuring techniques, so that small temperature gradients (which necessarily produce correspondingly small concentration gradients) may be used in the diffusion cell. The Soret coefficient is often strongly temperature dependent, and one wishes to identify the measured coefficient with a definite temperature; furthermore, the use of the smallest practicable temperature difference across the sainple cell iiiiniinizes convective disturbances, which give rise to serious errors if not adequately suppressed. Bryngdahl and co-workersg~ lo have recently devised a new kind of interferometer which is particularly convenient for the study of diffusion, electrophoresis, sedimentation, heat transfer, and similar phenomena. In the present paper, we demonstrate its application to the problem of nieasuring Soret coefficients and establish the conditions of operation which maxiniize its sensitivity for this purpose. The apparatus is shown schematically i n Figure 1. A detailed analysis of the princhiples of its operation is The Journal of Physical Chemistrg
available in Bryngdahl’s papers; we give here only a sufficient description to make intelligible our subsequent theoretical development and experimental results. Collimated monochromatic light illuminates the whole height a of the diffusion cell C, in which a refractive index profile is shown. L, and Lz are lenses which demagnify the shaped wave front emerging from C. The beam is then plane-polarized horizontally by passage through the polarizer P. The first Savart plate SI is positioned normal to the optic axis and oriented so that the angle between its principal planes is bisected by the plane of polarization of P,; it produces a vertical shearing of the wave front. The resultant wave fronts are polarized at right angles to one another, are of equal intensity, and are displaced by a vertical distance b
=
+
4 2 €(NoZ - Ne2),/(No2 Ne2)
(1) Visiting Research Associate from Department of Physics, Chalmers University of Technology, Gothenburg, Sweden. Supported in part by a grant from Stiftelsen Blanceflor Boncompagni-Ludovisi, fodd Bildt. (2) (a) C. C. Tanner, Trans. Faraday Soc., 49, 611 (1953); (b) H. Korsching, Z.,Vaturforsch., loa, 242 (1955). (3) J. A. Bierlein. C. R. Finch, and H. E. Bowers, J . chim. phys.. 54, 872 (1957). (4) J. Chanu and J. Lenoble, ibid., 53, 309 (1956); 55, 743 (1958). (5) H. J. V. Tyrrell, J. G. Firth, and AT. Kennedy, ,J, Chem. Soc., 3432 (1961). (6) J. A. Bierlein, J . Chem. Phys.. 36, 2793 (1962). ( 7 ) J. G. Becsey and J. A. Bierlein. ibid., 41, 1853 (1964). (8) L. G. Longsworth in “The Structure of Electrolytic Solutions,” W.J. Hamer, Ed., John Wiley and Sons, Inc.. New Tork, N. T.,1959. (9) 0. Bryngdahl and S. Ljunggren, J . Phys. Chem., 64, 1264 (1960). (10) 0. Bryngdahl. J. Opt. Soc. A m . , 53, 571 (1963).
THERMAL DIFFUSION MEASUREMENTS BY WAVE-FRONT-SHEARING INTERFEROMETRY
Figure 1. Optical arrangement of interferomet,er.
where N o and N e are the principal refractive indices of the bifringent crystals (each of thickness E) comprising the Savart plate. A second plate Sz,turned 90" with respect to SI, is placed in the converging rays of the lens L,. This plate produces (apart from an identical vertical displacement of both wave fronts, which is of no consequence) a horizontal displacement of each incident ray, proportional to its angle of incidence and opposite in direction for opposite polarizations. The net effect is to introduce a m a l l angle, as viewed froiii above, between the sheared wave fronts. An analyzer A, crossed with P, makes visible the interference fringes in the image plane AI2. The lens L2 foriiis an image of the exit window of C in the plane MI, whence it is transferred to M2by La. The iriterferograiii at RI, consists of two sharply focused and overlapping images of C, with the fringes superimposed in the area of overlap. The extent of overlap (best expressed as the relative shear, s = b / a ) is deteriiiined by the telescopic power f2/jl of the leiis combination LILz and can be varied by selecting appropriate focal lengths. The over-all vertical magnification nz of the interferoiiieter is of course deteriiiiried by the focal lengths of all three lenses arid by the distance h which depends on their disposit ion ; the horizontal magnification is influenced by the sanie parameters and in addition by the characteristics of the plate Sz. An explicit examination of these relations need not concern us here. In the case that the test zone contains only linear gradients in refractive index (an optically homogeneous riiediuni being included as the special case of a null gradient), the interference pattern consists of uniforinly spaced linear fringes, oriented vertically ; the distance between fringtbs in the image plane corresponds to a path difference, of one wave length. If nonlinear gradients exist, theii the fringes take a form which approximates to the derivative curve of the refractive index, this approxiination becoming exact in the limit of vanishing shear. I t is the special case of vanishing shear which Bryngdahl has specially emphasized in his theoretical and experimental work. This procedure has the advantage of requiring no mathematical description of a
1017
refractive index distribution in order to effect a direct experimental determination of its forin, but this advantage is secured at the expense of a low sensilivity. When dealing with a process of known phenonienology, it is possible-as we shall show for the case of thermal diffusion-to optiinize the shear for highest sensitivity. The resulting interferograms, although they are finitedifference curves rather than true dcrivatives, can be readily interpreted to derive Soret coefficients.
Theory The Difusion Process. The refractive index
p in a binary mixture during therniodiffusional uniiiixirig develops in time and space according to the equation3
Here p o is the initial uniform refractive index; z' is the vertical coordinate, measured upward from the midplane of the cell; a is the cell height; u is the Soret coefficient; 7 is the temperature difference across the cell; t is the time since the start; '6 = a2/$D, wherc D is the isothernial diffusivity; K , = r b p j d T , where T means temperature; arid K1 = no (1 - no)bp/bn, where n is the mole fraction of the component of lesser molecular weight and nodenotes the initial coinposition. After prolonged separation, a linear gradient in p is established. If the temperature gradient is then removed and the mixture is theriiiostated at the mean temperature, the ensuing isothernial remixing proreeds according to1'
where we now understand t to be measured from the instant that remixing begins. Since the shearing iiiterferoiiicter does not record any linear gradients, we see that the essential difference between eq. 1 and 2 is only i n the sign of their transient terms. l+"rointhis we expect that the interferograms of the unniixing and reniixing processes will be Identical iiiirror images at equal lapsed times. Optzcal Analyszs. Sirice we postulate the interferometer to be focused on the exit window of the sample cell, we need to define the optical path length R(v)of any ray leaving the cell i n the horizoiital plane 1'. assuniing that all rays enter the sample at nornial iricidcricc The general problem of formulating path length as a fulictlon of focal plane has been studied by Svensson,lZand (11) G Meverhof and K Nachtigall J P o l y m r Scz 57, 227 (1962) (12) H Svens-on, O p t Acta, 1 , 25 (1954), bee hii eq 30
Volume 69. Virmber 3
March 196 i
SILASE. GUSTAFSSON, JULIUS G. BECSEY,AND JAMES A. BIERLEIN
1018
the particular expression we require follows directly from specialization of his equations
striking in the plane v. Comparison of the last two equations gives
AzR(v) =
l/&(i+
+ i-) - b i
=
+
‘/Zb(R’+
R’-)
The prime stands for differentiation with respect to v. Since p’(v) is everywhere small, we neglect terms higher than the second in the series expansion; to the same order of approximation, we ignore any variability of in the denominator of the second term. We have now to consider the problem of evaluating the path difference AR(v) between pairs of rays-one from each of the two sheared wave fronts-which lie in the same horizontal plane v after emergence from the first Savart plate. The incremental retardation alR(v) resulting purely from the shearing is clearly equal to
A1R(v) = R+ - R-
(4)
where the positive and negative subscripts mean that the associated function of v is to be evaluated for v l / z b and v -- ‘lzb, respectively. An additional path difference a2R(v)is introduced right in the Savart plate itself. To formulate azR(v),we note first that the path differences between two sheared rays, originating from the splitting of a single ray which falls on the plate at an angle of incidence i, is given by Bryngdahl13 as b sin i, or more simply as bi if i is small. Figure 2 shows
+
v-b
I
y - 8
Figure 2. Diflerential retardation of rays entering a Sawtrt plate at differing angles.
two rays incident a t angles i+and i- at levels h b / 2 distant from the plane v, each split into two emergent rays of differing retardation. We seek to find the difference y - @ = A,R(v). Since a - /3 = bit and y - 6 = bi-, it follows that AzR(v) a - 6 = b(i+ i-). The term a - 6 may be eliminated if we recognize that 1 / 2 j a P ) - I / ~ ( Y 6) = bi = ‘ / z ( a - b ) - A,R(u), where i is the angle of incidence of the ray
+
+
The Journal of Physical Chemistry
+
-
bR’
(5)
in which we use the fact that i = dR/dv = R’. The total path difference AR(v) is the sum of eq. 4 and 5 , and with the aid of eq. 2 we can write it in the explicit form
AR(v) =
8
-
--I
c7K1Le-t/B
a2
RS
2
cos
RV
TS
as
7-1
+ 8L3 - [a7Kle-”’
as
2 2
[sin
Ps
+
3110 RS
cos as - as] sin
ZRV -
a
(6)
where s is the constant b / a . Except for the equation of a scale factor to be considered presently, this expression defines the shape of each one of the identical fringes which appear in the final image plane If, of the interferometer. In the limit s -+ 0, AR(v) = sLdp/dv, and the interferogram faithfully duplicates the derivative curve of the refractive index. However, in the general case of appreciable shear, this is not true. Whereas the first term of eq. 6 is always a symmetrical (even) function of v, the second term is an odd function which does not vanish if s is finite. The presence of the odd term leads us to expect a skewness or tilt i n the interferogram, which should decay away as time passes. This prediction is well fulfilled experimentally, as we shall see later on. We consider now the practical problem of evaluating an interferogram. The fringe shape can be characterized experimentally by a series of distances y(v) which define the displacement, measured perpendicular to the v axis, between points at arbitrary v values and the minimum point at v = 0. To relate these data to the optical path lengths defined in eq. 6, we recall that the distance between any two of the parallel fringes (call it T ) corresponds to a path difference of one wave length A. Since T can be measured directly on the interferogram and X is known, the horizontal scale factor r/X is readily established. It follows that the desired relation is y(v) = (?/A) [AR(v)- AR(0)1. This expression is somewhat awkward when written out explicitly, as it retains the skew character inherent in eq. 6. We can arrive at a neater solution and considerably simplify the subsequent problems of curve fitting by a judicious choice of experimental procedure. Let us
+
(13) 0.Bryngdehl, Arkiz: F y s i k . 21, 289 (1962): see pp. 331-333.
THERMAL DIFFUSION MEASUBEMENTS BY WAVE-~+~ONT-SHEARING Iwwimxo~~imiu
measure the deviations of pairs of conjugate points synimetrirally disposed around u = 0, and then dral with the iiiean values y*(u) = '/i[y(u) y(-u)]. In t e r m of this parameter, eq. G transforms to
+
87 S'X
[
urKll,e-'~e sin T
sin - (1
TS
TS
[I'
cos - 2 2 2
-
1
- cos
];
2 (1
2
2
2
T
- cos 5 (1 - s)~]
2
TS
TS
cos 2 2 ~
-
1'
2
x
- s)?
-
(7)
To cast (7) into the most general possible form, we must renieniher that the fringes i n the iirterferogram exist only in the region of overlap of the two vertically displaced cell images. Hence the coordinate u in the interferogram never extends to the maximum values *a/2 which obtain in the cell, but is liiiiited between closer bounds + ( 1 - 4 4 2 because of the shearing. I t is therefore convenient to introduce a new vertical coordinate 7 = 2u/(l - s)a; the range of this paranieter is always - 1 5 9 5 1, independent of s. Rewrit ing eq. 7 with this generalized coordinate gives
TS us TS - cos -~ - -][1
TS ~
1019
The area being determined on the interferograiiis by numerical int,cgration, the Sorct coefficient follows directly froin analysis of its temporal regression. The Opliiiium Shcar. 11. is iinportant. to the experimenter to know what, value of s to use in order to attain a maxiinuni sensil.ivity. To answer this question, it is necessary to investigate t,he null values of by/& and bA*/bs. The algebra involved is tedious but straight-
t.mh
Ve
0.0
0.00
24.I
0.40
57.9
0.96
106.7
I.TI
176.1
2.92
(8)
This expression forms the basis for determining the Soret coefficient. Since s can be calculated right froin the cell-image overlap (see Figure 1) and all the other constant,s can be measured beforehand, n follows from the regression of y*(?) upon exp(-l/O) at any fixed 7. A multiplicity of estimates for u niay be obtained by repenting the regression analysis for several fixed values of 9 ; this affords a safeguard against gross mistakes in measurement or ealculation and improves the precision of the final estiniate by averaging out unavoidable experimental errors. The ultimate extension of this procedure is to use all possible values of ?. In this connect,ion, consider the area A*(v), included between the convex side of a fringe and the tangent a t v = 0 and further bounded at * v . The analytical expression is
in which the unit value of A*(?) corresponds to one-half the area between two undeviated fringes. In practice, it is more convenient to work with the area on the convex side of the fringe, c l o d by the chord through the points * v . This area is given by
Fiwre 3. Experimental interferograms. Isothermal remixing of 0.5 M aqueous cadmium sulfate from the thermodiRusionnl stendy state, under conditions specified in the first column of Table I.
SILASE. GUSTAFSSON, JULIUS G. BECSEY,AND JAMES A. BIERLEIN
1020
forward and we do not, reproduce it here. Some typical numerical results summarize the situation. for optimum-----
B-
7
0 5 10
z/*
A*
0 2833 0 2727
0 2864 0 2734
It is clear that a shear of about 0.28 optimizes both cases over the range of 7 which is of practical interest. Further, we find by numerical analysis that the sensitivity changes slowly with the shear in the neighborhood of the maximum. For exaniple, if s is allowed to vary f 2 0 % around 0.2833, the value of y*(l) will change less than 5%. In general, any shear in the range 0.20 < s < 0.35 will do nicely for thermal diffusion nieasurements; outside these limits, however, the sensitivity begins to drop off rapidly.
Experimental Tests Apparatus. The cell used in our experiments consists of a rectangular glass frame of optical quality glass, clamped between horizontal plates of silvercoated copper; a thin film of silicone grease provides a seal against leakage at the contact surfaces of glass and metal. A riuniber of frames of different heights are available to permit selection of a convenient relaxation time for the solution to be examined; each franie measures about 6 X 8 cni. in plan view, the longer dimension lymg along the optic axis. The metal-glass interfaces at the top arid bottom plates of the assembled cell are not sharply defined or of good optical quality, owing to urtavoidable extrusion of the sealant; for this
reason, we mask off these regions with a large slit (height H ) , mounted on the exit window arid centered on the midplane. Three circulating water baths, regulated to a maximum ripple of *0.003", permit the cell to be operated under a steady temperature gradient or to be therniostated a t an intermediate temperature. Multijunction thermocouples, inserted into wells in the plates, are used to measure temperatures a t the upper and lower boundaries of the cell during operation. In our optical system we use a 500-watt inercury arc lamp to illumiriate the slit of a colliniator with a focal length of about 1 in. The light is filtered to isolate X 5461 8. Our Savart plates are made of quartz, E = 10 mm. and b = 84.2 1; they were supplied by Crystal Optics, Chicago, Ill. Two different system have been used to perform the shear-determining demsgnification in front of the first Savart plate (LIL2in Figure 1). One is a combination of a 914-mm. lens and a three-power microscope objective; another iiiore flexible arrangement utilizes a compact telescope (Questar) of adjustable focal length, A 40-mm. lens between the Savart plates completes the optical train. Photographic registration of the interference fringes is accomplished on Kodak Tri-X 35-nim. film with an exposure of about 1 sec.
Results To test the adequacy of our theory, as well as the utility of the instrument, we have measured the Soret coefficients of a nuniber of aqueous electrolytes for which reliable data already exist. Figure 3 is a typical
Table I : Experimental D a t a ; Mean Temperature 25'
__-KI
L , cm.
.
deg.
2.004 0.456 X lOP 60.3 0.202 0.196 0.562 0.568
e, min. S
.M AgNOs--
Remixing Remixing 0.402 0.805 0.284 0.571 0.0160 7.86 1 ,987 2.121 1.277 x 10-6 21.4 76.1 0.204 0.191 0.507 0.587
7 -
4.0 M KCl----
Unmixing Remixing 0.798 0.738 - 0.0254 7.88 2,213 2.213 2.165 x 10-5 49.7 0.256 0,720
X 103, sec.--I, from Y*(7)
'4Y7)
x
103, sec. -l, from literature
See ref. 8. ~~
1.0
-
2.143
D,cm.z/sec.
u
Remixing
0.404 0.284 -0.0127 7.86
H ,cm.
7 u
----
M CdSOd----
Remixing
Kind of experiment a, em.
7,
0.5
See ref. 6.
5 , 0 7 f 0.09 4.96 f 0.03 5 . 0 2 f 0.02 ... 4.92" 5 . 0 8 f 0 .OP 4 . 9 2 f 0.07"
See ref. 5.
~~~
Tlir Jnicrnal of Physical Chemzstry
See ref. 4.
e
See ref. 7
4.38 f 0.05 4.46 f 0 . 0 4 4.46 i 0.06 ... 4.27" 4.37 f 0.01c 4 . 4 6 f 0.06"
1 . 3 7 f 0.01 1 . 3 6 f 0.01 1.38 f 0 . 0 1 ... 1.33" 1 . 2 6 f O.Md 1 . 3 1 f 0.01"
THERMAL DIFFUSIONMEASUREMENTS BY WAVE-FRONT-SHEARING INTERFEROMETRY
set of interferograms from one of our experiments; the skewness predicted by theory in the early stages is clearly evident. Table I summarizes the results quantitatively. The Soret coefficient is computed from y*(q) in every run, and from A*(?) in some runs to demonstrate the substantial agreement which exists. In the two silver nitrate measurements, different cell heights were used; the two potassium chloride runs show the consonance between an unmixing and a remixing experiment. The precisions indicated for 0 are the probable errors computed from the scatter of points about the best-fitting regression curve. I t can be seen that our data have good internal consistency and agree well with previous determinations reported in the literature. To the extent that small differences do occur, our values tend to be higher. The reason for this is very likely connected with the fact
1021
that the high sensitivity of the instrument enables us to use lower temperature differences across the cell than was customary in most of the older work. As a consequence, parasitic remixing effects are much reduced. In summary, the shearing interferonieter gidds a precision which equals or exceeds that of ariy other optical instrument thus far described in the literature for measuring Soret coefficients. Moreover, it has appealing advantages from the standpoint of convenience. I t is very easy to set up and adjust, and since there is no need to provide an external reference beani, as in Rayleigh interferometry, the exacting problems of constructing and using a twin cell are eliminated. Further, the labor involved in analyzing an interferogram , while not insignificant , is much less arduous and time consuming than that required to read out inforination presented in the form of diffraction patterns, as in Gouyfringe or image-displacement techniques.
Volume 69, Sumber 3 March 1965
