GEOMETRICAL EFFECTSOF LIGHTBENDING AND REFRACTION
Corrections of Schlieren Data. I.
2843
Geometrical Effects of
Light Bending and Refraction
by T. Foster Ford U.S . Naval Research Laboratory, Washington, D . C . 20600
and Edwin F. Ford 9605 N . V a n D o r n Street, AZezandria, Virginia
(Received February 26, 1964)
The bending of light in an ultracentrifuge or electrophoresis cell, combined with its refraction on passing through the second cell wall, results in tilting of schlieren patterns and consequent displacements of plots of the center line position against time. The displacement is small but well outside errors of measurement. Sedimentation coefficients and lag times, the time delay in separation of schlieren peaks from the meniscus, are affected. A geometrical optical analysis is presented for the two-collimating lens system used in the Spinco ultracentrifuge. Cell thickness is found to be an important factor in determining the schlieren center line displacement. When thick cells are used, the bending-refraction effect by itself might cause schlieren center lines to appear to separate from the meniscus even before centrifugation started. There is no effect on diffusion coefficients. The indicated corrections should not be used without also applying the equivalent level correction resulting from curvature of the light path in cells of finite thickness.
Introduction Ultracentrifugal sedimentation velocity studies on barium dinonylnaphthalenesulfonate, with a micellar weight in benzene of about 8500,’ developed special problems in the interpretation of schlieren patterns because the diffusion rate is large as compared with the sedimentation rate. The data, analyzed in the usual way, showed a distinct lag time in the separation of schlieren peaks from the meniscus. These lag times agreed well with theoretical lag times calculated by a formula given by Fujita and MacCoshamP2 However, Svensson has pointed out3 that since the path of light entering a liquid column normal to a refractive index gradient is curved, an additional correction of the position of the sedimentation peak is required, and his estimate indicates a reduction of the observed lag time to the extent of 20% in our case. It was, therefore, concluded that the Svensson curvature correction must be reviewed. If it were found valid, a search should be made for compensating errors. The review of the curvature correction is presented in
part I1 of this report. The compensating error was found to be introduced by geometrical effects of light bending in the liquid plus further deviations due to refraction of the deflected light as it passes through the second cell wall. For the particular barium dinonylnaphthalenesulfonate data which prompted this study, the effect is opposite in sign and nearly equal in magnitude to that introduced by curvature.
Results QualitativeEfects of Bending and Refraction. Figure 1 is a dimensionally distorted diagram of that part of the Spinco optical system between the cell and the plate, with the cylindrical lens omitted. It shows (1) T. F. Ford, S. Kaufman, and 0. D. Nichols, “Ultracentrifugal Studies of Barium Dinonylnaphthalenesulfonate-Benzene Systems. I. Sedimentation Velocity,” presented a t the Kendall Award Symposium, 147th National Meeting of the American Chemical Society, Philadelphia, Pa., April, 1964. (2) (a) H. Fujita and V. J. MacCosham, J . Chem. Phys., 30, 291 (1959) : (b) H. Fujita, “hlathematical Theory of Sedimentation Analysis,” Academic Press, New York, N. Y., 1959, p. 87. (3) H. Svensson, Kolloid-Z., 90, 141 (1940).
Volume 68, Number 10
October, 1964
2844
T. FOSTER FORD AND EDWIN F. FORD
I-
OF CELL
UPPER +-COLLIMATING LENS
I
3CHLlERN DIAPHRAGM -CAMERA LENS
1
/
_ _%OF OPTICAL SYSTEM ’
Figure 1. The principal part of the two-collimating lens schlieren optical system, omitting the cylindrical lens. inserted diagram is a detail showing the path of a refractive index gradient-deflected ray through the cell only.
the paths of four rays. Two of these just graze the holes in the cell counterbalance and produce the upper and lower vacuum reference lines on the plate. These rays intersect the cell centerline a t the points R1 and Rz. One of the other two rays passes through the cell without any bending and intersects the cell center line a t the point b; the last ray is bent in its passage through the cell, as would occur if a refractive index gradient were present. We consider first the two reference rays and the one undeflected ray through the cell. The camera is focused on the cell center line, as is usual practice, and it records the virtual levels Rl,o, Rz,o, and bo for these , rays. The respective image levels are R ’ I , ~ R’*,o, and b’o. The image distances between these camerarecorded points and the optical center line are O’-R’j ,o, O’-R’2 O , and O’-b’o, respectively. From similar triangles it is seen that these distances are in proportion to the real distances, 0-R,, O-Rz, and 0-b (cf. Svensson, ref. 3). Therefore, if there were no bending, all levels in the cell would be proportionally recorded T h e Journal of Physical Chemistry
--
The
and would be correctly calculated using a proper magnification factor. We now consider the ray which is bent. Reference must be made to the inserted diagram of Fig. 1, which shows the light path for this ray through the cell only. Here, for simplicity, the refractive index of the liquid is presumed to be equal to the refractive index of the material of the cell wall, nw,as is nearly the case when the cell liquid is benzene. Thus, in this case, the only bending due to refraction occurs as the ray exits the cell wall, and the deflection inside the cell wall is equal to the deflection of the ray on exiting the liquid. To a first approximation, the path of the ray through the liquid is a p a r a b ~ l a . ~Therefore, the backward extension of the exit tangent intersects the center line of the cell on the entry level, or a t b. Referring now again to the main figure, it is seen that in consequence of the total deflection of this ray the (4) 0. Lamm, Nova Acta Reg. SOC.Sci. Upalhiensis, 4 , 10 (1937); T. Svedberg and K. 0. Pedersen, “The Ultracentrifuge,” The Clarendon Press, Oxford, 1940, p. 257 ff.
GEOMETRICAL EFFECTSOF LIGHTBENDING ASD REFRACTION
camera records the image position b’,, therefore the virtual position b,. The difference between b’o and b’&is the camera-recorded displacement of the entry level due to bending and refraction, and bo-b, is the absolute virtual displacement. Figure 1 was drawn by geometrical construction. Other, similar constructions drawn to scale show that the displacement may vary greatly with cell thickness. For thick cells it is reversed, so that b, appears to be above bo rather than below it. Formulation. Thc correction for errors due to bending and t o refraction can be formulated quantitatively. We refer to Fig. 2 and also to Fig. 3, which is an enlarged detail of Fig. 2. In these figures:
2845
in the cell; F is the focal length of the upper colliinating lens, i e . , the optical lever arm; G is the distance from the cell wall, on the camera side, to the principal plane of the upper collimating lens; W is the thickness of the cell wall; and C = of the inside thickness of the cell. We also introduce the term 6, to designate the required refraction correction of x, in centrifuges, the distance from the center of rotation. We designate the refractive index of the liquid by nl and the refractive index of the material composing the cell walls by n,. We now note
G F
6/= -6
B = abo = 6
*-
6’
- 6,
- 6oell
T T + # F; ab, = (6 + B ) j j
where Figure 2 . Diagram illustrating the bending and refraction correction for deflected rays.
UPPER COLLIMATING LENS
‘
T=C+W+G+F
(5)
Therefore 6, = abo - ab, =
s + + -T - 6 -T-
F
F
B -T F
Substituting (4) for B in (e), and remembering that T = C W G F , we find
+ + +
It is seen from eq. 7 that (1) 6, is positive when the Figure 3. Detail of Fig. 2. Here 6, is the bending and refraction correction.
4 is the vertical distance between the entrance plane of a particular light ray, or sheet, and the center line of the optical system; S is the observed vertical deviation of the light sheet a t the schlieren diaphragm; 6’ is the deviation of the light sheet between the camera side of the cell wall and the principal plane of the upper colliinsting lens; 6, is the deviation of the light sheet in the cell wall; B c e l l is the deviation of the light sheet
combination of terms on the right is less than 6. I n this case the cell level in question appears to the camera to be too far from the meniscus, or from the center of rotation, The quantity 6, is therefore to be sublracted from the apparent x-value as recorded by the cameraschlieren lens system; ( 2 ) 6, increases as the ratio T / F is increased. We consider 6 to be that portion of the total deviation due to the concentration gradient in the cell. I n the absence of density gradients, preformed gradients, and cell distortion, as in electrophoresis, 6 is the total deviation. In the ultracentrifuge it is that deviation given by the heights of schlieren ordinates above the Volume 68, Number I O
October, 1864
T. FOSTER FORDA N D EDWIN F. FORD
2846
base line. The absolute value of this deviation may be calculated from the heights of the ordinates, the tracing/plate magnification factor, the cylindrical lens magnification factor, and the bar angle. I n the Spinco optical system the maximum possible deviation giving complete resolution of patterns is set by obstacles in the light path, e.g., the wire clips on the mirror mounting, and it may therefore be calculated from measurements on the apparatus with some accuracy. On a schlieren pattern maximum deviation is indicated by the appearance of a Toepler schlieren center line due to the extreme light sheets hitting the obstruction responsible. This maximum resolvable peak height once determined and the corresponding maximum deviation at the schlieren diaphragm, 6, known, deviations at other points on the pattern or on other patterns may be calculated by proportion, taking proper account of bar angles. Numerical Calculations. Equation 7 is cumbersome because of the many symbols. It is better to make the calculations by steps. The following example, for a water system, is offered. The dimensions to be used are those for our apparatus. A 1.2-cm. cell is assumed. The refractive index of the fused quartz walls is taken to be 1.467 for X = 0.434 p at 25’; the refractive index of the solution, 1.34. We use the diniensions: ‘/z of the cell thickness (C), 0.600 cm.; wall thickness ( W ) , 0.500 cm.; wall to collimating lens (G), 4.531 cm.; optical arm ( F ) , 48.504 cm. These dimensions give a total of 64.135 cm. Referring directly to Fig. 2 and 3, without returning to the equations just derived, and using the preceding values, we find: 6’ = 0.077458; 6, = 0.0058268; 6 c e l ~ = 0.0076548; B = y5 - 0.090938; abo = 6 1.09625$; and ab, = (6 B)(1.09625); whence 6, = abo - ab, = 0.003436; and 6,/6rel~ = 0.448. For our apparatus, the maximum total deviation a t the schlieren diaphragm before appearance of a center line, as calculated from tracings using the cylindrical lens magnification factor, is about 1.80 cm. Actual measurements give about the same value. Of this 1.80 em., 1.43 cin. is above the base line (at 59,780 r.p.ni.). If we use this value, 1.43 em., for 6 we obtain: 6cel, = (0.007654)(1.43) = 0.01094 cm.; and 6, = (0.00343) (1.43) = 0.00490 cni. This absolute value for a, the displacement of the camera-recorded 2-value due to refraction, is about five times the possible precision of measurements. Actually, schlieren patterns are resolvable at deviations as much as 20% greater than that used above, and schlieren center lines may be found, by bisecting the Toepler schlieren line, a t deviations as much as
+
The Journal of Physical Chemistry
1 0 0 ~ greater. o For such unresolved patterns the peak heights and therefore the deviations may be found by extrapolation of plots of the resolved peak heights against 1/deiapsed time. Thus it is possible to calculate 6, for such unresolved patterns. For these, then, we would find for the diniensions cited, G c e i i 4 0.02188 cm., and 6, = 0.00980 cm. These values were calculated from actual data obtained on a bovine serum albumin solution, l.22%, 59,780 r.p.m., at 25’. The pattern giving 6, = 0.00980 cm. was obtained 10.6 niin. after starting the rotor; the pattern giving 6, = 0 00490 cm., mas obtained 50 min. after starting the rotor. Resolvable patterns were obtained at intervals up to 154 min. after starting the rotor. Correlations with Cell Thickness. Calculations such as those outlined above were made for all the cell thicknesses commonly employed in the Spinco ultracentrifuge, and also for an infinitely thin cell and for a cell completely filling the space betwcen the collimating lenses. The other diinensions used were those given in the preceding section. These calculations were made for both benzene and water solutions. The refractive index of benzene is 1.5200, for X = 0.434 p , a t 2Z0,5 but to simplify the calculations it was assumed to be the same as that for quartz. The values for 6,, max and 62/60ell are plotted against cell thickness up to 4cm. inFig. 4. (For the hypothetical cell filling the space between the collimating, lenses, 10.26 cm., 6,/6cell = -0.407 for benzene and -0.277 for water, and the &-values are -0.0347 cm. and -0.0258 cm., respectively,) I n the calculations of 6,,
+
I 0
,
1
10
20
CELL
TtMNiSS
,CMl
30
1
T
40
Figure 4. The variation of ti,, and of the ratio of 6= t o the cell deviation, &,ll, with cell thickness, for water and benzene systems. (5) D. E. Gray, Coordinating Editor, “American Institute of Physics Handbook,” McGraw-Hill Book Co., Inc., New I-ork, N. Y . , 1957, pp. 6-18,
GEOMETRICAL EFFECTS OF LIGHTBENDING AND REFRACTION
the deviation a t the schlieren diaphragm was assumed to be 1.43 cm. which, as noted above, is not the absolute maximum usable deviation. Distortion of Guussian Patterns. Figure 5 is a representation of the nature of the distortion of gausshn patterns due to light bending and refraction alone. The data for drawing this figure were obtained in tbe following manner. We have chosen to use the 1.5-mm., 1.20-cm., and the hypothetical (10.26-cni.) cell, for which the maximum &-values for completely resolvable patterns, for water solutiom, are, respectively, f0.00846, +0.00490, and -0.02584 cm. We assume that the actual dc/dx vs. x distribution in the cell is gaussian. On the gaussian scale,6 the peak of the curve is a t 1.12838. At gaussian x-values to the left and right of the center line of a gaussian curve the corresponding de/dx, or H‘, values are given in tables of the error function.’ For successive gaussian X-values, we can, therefore, calculate the ordinate, H i , as a fraction of the maximum peak height. The S,-values will be proportional to these fractions. It remains to convert these &-values to the gaussian x-scale. For this we take our particular experimental bovine serum albumin pattern for which, in a 1.20-cm. cell, 6, == 0.00490 cm. For this pattern the actual distance between the two sides, 2u, a t the inflection level is 0.09035 em. in absolute units, and it is 1.4142 on the gaussian X-scale. Therefore, our 6,-values, proportionated t o the ordinates, must be multiplied by the factor 1.4142/ 0.09035, or 15.65. For any oth.er cell thickness the factor is the same, because the width of the actua,l
INFLECTION
X
2
LEVEL
3
-
+4
Figure 5. Theoretical distorted camera-recorded schlieren patterns compared with the gaussian pattern presumed t o exist in the cell, for various cell thicknesses.
2847
gaussian distribution in the cell is the same. The gaussian S,-values calculated in this way for several levels were algebraically added to the gaussian x-values for the same levels, thus obtaining the patterns shown in Fig. 5. The theoretical gaussian curve is included for reference. These distorted patterns are those that would be recorded by the camera if the real distribution in the cell were gaussian, and if the refraction correction were the only correction to be considered. It is seen (Fig. 5) that the patterns for the 1.5-mm. cell and the 1.20-cni. cell are tilted to the right, and that the pattern for the hypothetical 10.26-em. is tilted to the left. For an infinitely thin cell, the pattern is tilted farther to the right than for the 1.5-mni. cell in the ratio 0.00896/0.00846, or 1.06 (cf. Fig. 4). For each of these examples succeeding pictures in a time series would show decreasing displacements in proportion to the decreasing peak heights. The schlieren center lines obtained by bisecting the patterns (Fig. 5) are straight lines, but they are not vertical. Therefore, a level of reference must be chosen, and x-values measured at this level. The apparent inflection level, for which the ordinate is 0.6065 times the peak height, is to be preferred for two reasons: (1) patterns are usually best defined in this region and precision of measurement greatest; (2) gaussian curves are almost linear a t this level and for appreciable distances on both sides, which simplifies analytical calculations (cf. part I1 of this report).
Discussion The bending and refraction correction of schlieren center lines has a very great effect on the time intercept of z-time plots and therefore on lag times. For thick cells such plots might intercept the meniscus apparently before the centrifuge was started, and the observed lag times could be negative. The refraction correction by itself also has an effect on the slopes of 2-time plots and therefore on s-values calculated from them. For the particular bovine serum albumin experiment cited the refraction-corrected xinfleotion-time plot gives an s-value about 0.2% greater than that given by the uncorrected plot. For a 1.5-mm. cell this correction would be almost doubled. If the center line 2-values a t the peaks of the curves were used these corrections would be multi(6) In this report, and in the second part also, dimensions are frequently expressed in “gaussian units.” In gaussian units the maximum ordinate of the normal distribution curve is always 1.12838 ( 2 / & ) , and the distance across the pattern at the inflection level, or at 0.6065 times the ped< height, is always 1.4142 (d5). ( 7 ) National Bureau of Standards, “Tables of t,he Error Function and its Derivative,” Applied Mathematics Series, 41; The Superintendent of Dowments, U. 8. Government Printing Office, Washington 25, D. C., 1954.
(l/da
Volume 68, Number IO
October, 1961
2848
plied by 1/0.6065, or 1.65, and for a 1.5-mm. cell the correction of s would be about +0.6%. It is emphasized, however, that if the 6, correction be used it must not be used alone (cf. part I1 of this report). The refraction correction has no effect on u-values, because a t each level the patterns are displaced equally on the left and right and in the same direction. Peak heights are affected, but the areas under the patterns are also changed and in the same proportion. Therefore, there is no refraction effect on the values of diffusion constants calculated either from u-values or from peak heights and areas. I n the calculations given here, effects of centrifugal pressure gradients, preformed gradients, and cell distortion are presumed to be constant throughout any particular experiment and, therefore, are presumed to have no bearing on the &-value to be used. This &-value is calculated from the deviation caused by the concentrationgradient, that is, from the height of the schlieren ordinate above the base line. When patterns are corrected to horizontal base lines as is common practice, all of these effects are combined as one and
The Journal of Physical Chemistry
T. FOSTER FORD ASD EDWIN F. FORD
we have left a pattern describing the concentration gradient only. I n electrophoresis, where density gradients and cell distortion are presumably absent, these matters need not he considered. The numerical results given are for the Spinco ultracentrifuge optical system. For other optical systems, e.g., electrophoresis, other dimensions would be required. The general formulation given would apply, however. No account has been taken of nonhorizontal light entry, as would be encountered in a pressure-bowed cell or in a wedge-shaped cell or for optical systems employing only one collimating lens. If the light entry angle were constant in the x-direction, the resultant bending would be constant and would, therefore, be eliminated by the use of a blank-run base line. For any particular optical system and a particular liquid medium, the refraction correction becomes zero a t a definite cell thickness (cf. Fig. 4). For water systems in the Spinco apparatus, this occurs at a cell thickness of 2.7 cm.; for a benzene system, a t 2.08 cm.
