IRWIN H. BILLICK.4ND ROBERTJ. BOWEN
4024
values we propose to obtain a numerical solution of the Poisson-Boltzmann equation for a cylindrical
capillary and compare it with the approximate solution.
Application of a High Intensity, Multislit Rayleigh Interferometer to Sedimentation Studies
by Irwin H. Billick and Robert J. Bowen ,MaeromoEecuZes Section, National Bureau of Standards, Washington. D . C.
(Received March 8, 1965)
I n order to obtain a high light intensity interferometer for sedimentation studies, the multislit system of Svensson has been used with the Spinco Model E ultracentrifuge. Other advantages of the multislit system, in addition to greatly increased light intensity, are (1) the simultaneous recording of the schlieren pattern of the refractive index gradient as well as the integral interference curves and (2) the interference fringes, extending over about 150 fringes, which make it possible to follow one fringe across the entire cell, even for concentrated solutions. Examples are given of the application of this optical system to high speed velocity sedimentation and low speed equilibrium sedimentation of polystyrene in cyclohexane. I n the case of the velocity sedimentation, studies were made over a concentration range below that normally used with schlieren optics. Accurate values of the concentration were obtained by suitable extrapolation to zero time of sedimentation, and values of the sedimentation coefficient and pressure dependence parameter could be calculated from the extrapolation data. For the particular systems studied, the existence of the plateau region was observed, within experimental error, contrary to the prediction of the theory of sedimentation of pressure-dependent solutions.
Introduction The use of interference optics for the measurement of concentration in a rotating ultracentrifuge cell has several advantages over the more commonly used optical systems. Foremost of these is its superiority over other optical methods for the accurate determination of concentrations.’ In addition, since the sensitivity of the interferometer is greater than that of the more commonly used schlieren system, it is very important for studies of materials that exhibit large concentration dependence. While light absorption optics possesses even greater sensitivity, it is unfortunate that many macromolecules of interest do not have a chromophoric group in the molecule. I n spite of its advanT h e Journal 0.f Physical Chemistry
tages, the application of the interferometer for producing ultracentrifuge patterns has been slow2; what acceptance it has had, has been mainly for sedimentation equilibrium experiments or low speed velocity studies. I t has been observed that at the high speeds normally used for sedimentation velocity studies, the fringes produced by the Rayleigh interferometer become blurred and eventually disappear. Several possible steps which can be taken to improve the quality of the fringes have been suggested in the literature. For (1) H. K. Schachman in “Ultracentrifugal Analysis in Theory and Experiment,” J. W. Williams, Ed., Academic Press Inc., New York, N. Y.,1963. (2) H. K. Schachman, Biochemistry, 2 , 887 (1963).
APPLICATIONOF RAYLEIGH INTERFEROMETER TO SEDIMENTATION STUDIES
example, replacement of the quartz cell windows with sapphire reduces some of the fringe distortion. A second procedure for improving the quality of the fringes is to reduce the width of the light source ~ l i t . ~ J The dimension of the slit width depends upon the characteristics of the individual optical system; however, calculations which may be considered typical for the interference optical system in the Spinco Model E ultracentrifuge have been carried out by LaBar and B a l d ~ i n who , ~ found that the maximum permissible source slit width for their system was approximately 0.02 mm. Decreasing the width of the source slit improves the quality of the fringes considerably and is independent of the experimental variables, such as material studied and centrifuge speed. However, it has been observed that the widths of the fixed apertures are not independent of these variables and that their niodification also can improve fringe q ~ a l i t y . ~These , ~ apertures consist of two parallel slits and are usually fastened on the upper schlieren lens h 0 1 d e r . ~ ~If~ the widths of these slits are too large, fringe distortion results which is a function of the speed of the centrifuge. In the first place, blurring results because too much of the rotating cell is “seen” at given times. Secondly, when large refractive index gradients are produced in the cell, the resulting interference patterns are blurred or annihilated.* Obviously, refractive index gradients, even with solvent alone, are produced by hydrostatic pressure, which is a function of speed, and are greater in the case of compressible organic solvents than in the case of relatively incompressible water. It was found2 that decreasing the width of both the slits results in an improvement in fringe quality. -4 typical modification has been a reduction of width froin 0.75 to 0.240.36 mm.* Experiments were tried which applied all of the previous recommendations to the study of the velocity sedimentation of polystyrene in cyclohexane but met with little success. It was found that, in addition to the first two recommendations, it was necessary to reduce the upper lens slits to 0.20 mni. before good quality fringes were obtained. When both the source and the upper aperture slits have been narrowed to the point where good quality fringes are observed, the intensity of the light that impinges upon the photographic film is so low that exposure times become too long for sedimentation velocity studies where a rapidly moving boundary is present. Some increase in light intensity can be obtained by redesigning the cell window holder,*but the increase is not as much as would be desirable. The problem, I herefore, is to increase the intensity
4025
of light passing through the optical system. This, of course, can be accomplished by replacing the light source with one of greater intensity, but a sinipler method exists. A design for a high intensity Rayleigh interferometer has been given by Svensson’ and has been applied by him to diffusion studies. In the present work, we have applied Svensson’s method to the Spinco Model E ultracentrifuge. The only modification to the present commercial optical system is the replacement of the single light source slit by a multiple slit having specified dimensions. In the single slit system, a series of fringes is produced within a central diffraction band. When niultiple slits are used, an interferogram will be produced by each slit, and, if the spacings between the slits are sufficiently close together, the individual interferograms will overlap. If the separation is chosen correctly, the fringes from the individual patterns can be made to coincide, and the resulting niultifringe pattern will now be much brighter than that resulting from a single slit. In addition, the resulting interferogram has the property that the fringes cover the entire photographic plate in the direction normal to the radial direction making it possible to follow asingle fringe throughout the entire cell, even for a concentrated solution.
Experimental Section Svensson showed? that, for the type of Rayleigh interferometer presently used in the Spinco 1Iodel E, the requirement for fringe coincidence is given by
where e is the grating constant, ie., the slit separation, f is the focal length of the lower collimating lens, d is the separation of the slits in the Rayleigh mask, and X is the wave length of light. We have chosen values of the parameters, given hi eq. 1, so that they are adaptable to available instruments. The fixed upper aperture was the symmetrical type3 and also served as the Rayleigh mask. Several pairs of slits having variations in widths from 0.33 to 0.15 mm. were machined from brass. The separations of the centers of these slits were measured on a inicrocomparator and were found to have an average value of (3) E. G. Richards and H. K. Schachman, J . Phys. Chem., 6 3 , 1578 (1959). (4) F. E. LaBar and R. L. Baldwin, ibid., 66, 1952 (1962).
(5) F. A. Jenkins and H. E. White, “Fundamentals of Optics,” hIcGraw-Hill Book Co., Inc., New York, N. S., 1950, p. 314. (6) Spinco Division, Beckman Instruments Inc., Technical Bulletin No. 6001-B, 1958. (7) H. Svensson, Bcta Chem. Scand., 5 , 1301 (1951).
Volume 69, Number 11 .\‘ovember
1966
4026
IRWIN H. BILLICKAXD ROBERT J. BOWEX
*
d = 0.397 0.001 mm. The various width slits were all tried in the optical system, and the pair having a width of 0.20 mm. was found to be the most suitable for general use. A Baird-Atomic interference filter having a peak wave length of h 5460 8.and a half band was used although acceptable results width of 84 were obtained using the Wratten 77 or 16 filters. The focal length of the lower collimating lens was measured by the Refractometry Section of the Sational Bureau of Standards and found to have an equivalent focal length of 587.75 nini., as contrasted to the nominal focal length of 590.55 mm. stated by the supplier. The value of the grating constant was therefore e = 0.0808 mm. The value of the niaximuni source slit width, PP',4 for our system can be evaluated from the above data accordirig to
H.
The value of PP' was found to be 0.0202 mm.; however, a value of 0.01 mni. was used in order to improve the image quality still further. The light source in the Model E is a high-pressure mercury vapor lamp, Type AH-6. It has a usable length of 1.90 cni., and a grating of 201 slits with the above parameters was used so that the entire length of the source rvould be utilized. Rather than have only one grating made which would be used directly, a negative was manufactured to our specifications by the Bausch and Lomb Co. The master negative was constructed by ruling slits in a deposit of Inconel on glass. The dimensions of the negative were such that the positive slit dirnerisions mere 0.010 f 0.0025 mm. by 10.0 =t 0.1 mm. The spacing accuracy between any ad0.0025 mm. Working gratings were jacent slits is obtained by contact exposure of the master negative on 3 X 2 in. Kodak High Resolution photographic plates. -4t first, an attempt was made to place the grating directly over the light source.' However, even though water cooling was used, the grating was destroyed by heat in a matter of seconds. The final arrangement was to place the grating on top of the dust cover of the light source, separated from the source itself by a piece of heat-reflecting glass (Bausch and Lomb Type Xo. 9012). The light filter, in turn, was mounted on top of the grating. The alignment of the optical system was carried out following the procedure recommended by Gropper.* For initial positioning of the height of the light source, the multiple slit was replaced by a single slit, made by exposing a photographic plate over which a thin wire had been stretched. The slit was positioned emulsion
*
The Journal of Physical Chemistry
side up and the height of the source adjusted so that the slit image on the photographic plate was located in the focal plane of the lower collimating lens. The z/s plane was chosen for the plane of focus of the camera lens on the centrifuge cells to eliminate Wiener skewness. As an aid in some steps of the alignment a Polaroid back was mounted over a hole in the back of the camera and was used where focusing on the plane of the photographic plate was not required. For the current investigation, studies were made of the sedimentation of S.B.S. Standard Polystyrene Sample S o . 705 in cyclohexane at 35". A total of four concentrations, ranging from 0.05 X to 0.3 X lo-* g./cc., was investigated using a 12-mm. thick, aluminum-filled epoxy, double-sector centerpiece. Since it was desirable to have equal levels in both the solvent and solution compartments, the centerpiece was modified so that a connecting channel existed at the extreme top and bottom of the sectors. Khen the cell was filled, the amount of solution used was such that the solution was at a slightly higher level than the solvent. At a fairly low speed, the levels would equalize, and a synthetic boundary would be formed, which was always close enough to the cell bottom so that it would sediment to the bottom and no longer be visible in the pattern. The same cell was used, without disassembling, for all concentrations, and a blank was run using solvent in both sectors. All rims reported here were made at 50,740 r.p.m.; however, runs made at speeds up to 60,000 r.p.m. exhibited satisfactory fringe patterns. Photographs were taken on Iiodak II-G plates, with exposure times varying between 15 and 2.5 see. I n order to enhance the contrast , plates were underexposed and overdeveloped in D-19 developer. To increase contrast of the fringes further, contact prints were made on Kodak Metallographic plates, and the fringes were then read on a two-dimensional comparator. The comparator is ac* curate to i1 1.1. Figures 1 and 2 illustrate typical patterns obtained using this optical system. The pattern of Figure 1 was obtained from a 0.3 X 10-2-g.,/cc. solution 80 min. after time zero a t 50,740 r.p.m. Figure 2 is a photograph of the solvent pattern taken ai the same speed. One of the advantages of this inultislit system is the extent of the pattern, which permits the same fringe to be followed throughout the entire cells7 This was, therefore, the procedure used when the fringes were measured. Measurements were taken beginning as close as possible to the meniscus and extending as near ( 8 ) L. Gropper, Anal. Biochem., 7, 401 (1964).
(9) H. Svensson, O p t . Acta, 1, 25 (1954).
APPLICATION OF RAYLEIGH INTERFEROMETER TO SEDIMENTATION STUDIES
4027
t .L Figure 1. Photagmph of iuterferewe pittern given by a 0.3 X lO-Lg,/cr, soltilion 01 pulyslyreiie i i i cyrlohernne Using milltislit optics. Pattern produced niter ,SO min. of centrifugation at 50,740 r.p.m.
Figure 2. Plrotogmph of d v c n t fringe. obtsincd with multislit optics nt 50,ilO r.p.m.
t o cell bottom as possible, recording both the displacemeht of the fringe and its position. The interval b e tween measurements in the radial direction varied, with smaller intervals taken in the boundary region. The particular system described produces a pattern containing about 150 fringes. Measurements of the
Figure 3. Patterns produced by two-cell opernlion nt S563 r.p.m. Solutions are polystyrene i n rydohexune: upper solution concentration 0.398 X 10-2 g./cc.; lower solution concentration 0.208 X 10-1 g./cc. Photgraph shows both interference and schlieren patterns as described in text.
spacings between 20 successive fringes showed that the spacing on the photographic plate was 0.269 mm. with a standard deviation of 0.008 mm. Both the fringe spacing and the standard deviation are of the same magnitude as a comparable single-slit system.I0 Measurements of the blank showed that the fringes were not linear; however, the blank did not appear to be speed dependent and was reproducible, as has been reported by others.I0 This particular optical system is also adaptable to multicell operation, by proper use of false bottoms and masking. Figure 3 illustrates a two-cell equilibrium run of polystyrene in cyclohexane. Results obtained from these experiments are not reported here, but the figure is included as an example of the utility of the optical system. It will be noticed that, here again, it is possible to follow one fringe throughout the entire pattern, so that fringe counting and calculation of fmctional fringes' are not necessary. Figure 3 shows an additional advantage of this particular optical system, namely, its ability to produce the schlieren pattern of the refractive index gradient simultaneously with the integral interference curve.' To produce this particular pattern a slit 0.20 mm. wide was superimposed on the grating so that the slits on the grating were per(IO) D. A. Yphnntis, Biahhemiairu. 3,297 (1964).
Volume 69. N u h r 11 h'm&r
1866
IRWIN H. BILLICKAND ROBERTJ. BOWEN
4028
pendicular to the second slit, and an exposure was made. It was necessary, however, to increase the exposure time by a factor of 4. Figure 3 was taken a t 5563 r.p.m. and a phase plate angle of 75”. Preliminary processing of the data from the velocity experiments involved conversion of plate position to radial position followed by subtraction of the base-line reading a t constant radius. The corrected data were then numerically differentiated twice with respect to radial distance and the results inspected. The boundary was chosen as the region over which the fringe displacement showed a monotonic increase with distance from the center of rotation. In theory,l1,l2for a system in which the solvent and solute show a large pressure dependence, there is continuous increase in concentration from the solvent side, through the boundary region, and all the way to the cell bottom; ie., the “plateau” does not exist. At the velocity and concentrations used in this study, calculations according to the theory of Fujita12 showed that this effect should be measurable within the experimental error of the niultislit optical system. However, none of the data indicates this phenomenon, and, for all the systems reported, a “plateau” region was assumed to exist within experimental error. Therefore, the concentration Iiahange across the boundary was calculated from the difference in fringe height between an average of first and last points surrounding the boundary. Three methods were used to locate the average boundary position, viz., the second moment,la the radial position at which the concentration is equal to one-half the plateau concentration, and the radial position at which the second derivative of the concentration curve with respect to radial distance is equal to zero ( L e . , peak maximum). Analysis of the data did not show a significant difference for the values of the boundary position calculated by the above methods. Therefore, only the results based on the radial position at one-half the plateau concentration are given. Sedimentation coefficients were calculated according to previously described methods14 in which corrections were made for pressure effects and radial dilution. ,411 calculations were carried out using an IBM 7094 electronic computer with programs written in FORTRAS.
Results If the assumption that a plateau region exists is valid, the concentration change across the boundary, c, is given by12r15r16 eq. 3a and b, where y = ( r / r o ) z ,r being
The Journal of Physical Chemistry
c
1
- = -
Y2
exp(1
+ m)7
the radial position of the boundary and ro the meniscus position, r = 2 w 2 t , w being the angular velocity, t the time, SO the sedimentation coefficient at 1 atm. pressure, co the concentration a t t = 0, and m the pressure dependence parameter as defined in ref. 12. These equations suggest plots from which values of co, so, and m may be obtained from the measured values of c and y. Figure 4 shows data plotted as l/cy us. y - 1, and Figure 5 shows plots of In cy2 vs. u2t. A somewhat similar treatment of the data, obtained through the use of schlieren optics but a t concentrations higher than those reported here, has been reported previously.‘’ The results for the concentrations a t zero time and y = 1 given in Table I were obtained by least-squares analysis of the data according to eq. 3a and b. Columns 1 and 2 give the calculated zero time concentrations using a value of dn/dc for polystyrene in cyclocc./g.18 The concentrations hexane of 0.1707 X
Table I: Concentration Determined Using Multislit Optics CO
co X 102, g./cc.
(calcd.), fringes
0,051 0.100 0.199 0.297
1.90 3.74 7.44 11.10
a
co
co
0.96 0.99 1.01 0.98
0.08 0.03 -0.06 0.22
(measd.)/ (calcd.) (measd.),a (measd.)? co co (measd.), fringes fringes (calcd.) fringes co
co
1.82 3.71 7.49 10.88
1.83 3.72 7.50 10.88
Determined from eq. 3a.
‘ Determined from eq. 3b.
obtained by extrapolation of eq. 3a and b are given in columns 3 and 4,respectively. The ratios of co (measured) to co (calculated) are given in column 5 and, with the exception of the lowest concentration, show the good accuracy which is obtainable. The values of the differences between co (calculated) and co (measured) are given in column 6 and are, for the most part, of the order of magnitude which one would expect from the precision of measurement of the fringes. The agreement between eo (calculated) and eo (measured) indi~~
~
~~~~~
~~~
(11) A. F. V. Eriksson, Acta Chem. Scand., 10, 360 (1956). (12) H.Fujita, J . Am. Chem. Soc., 7 8 , 3898 (1956). (13) R. J. Goldberg, J . Phys. Chem., 5 7 , 194 (1953). (14) I. H.Billick, ibid., 66, 1941 (1962). (15) 31. Wales, J . Am. Chem. Soc., 8 1 , 4758 (1959). (16) I. H.Billick, J . Polymer Sci., 62, 167 (1962). (17) J. E. Blair and J. M’. Williams, J . Phys. Chem., 68, 161 (1964). (18) J. H.O’Mara and D. LMcIntyre, ibid., 63, 1435 (1959).
APPLICATION OF RAYLEIGH INTERFEROMETER TO SEDIMENTATIOX STUDIES
4029
2.8
2.4
+r 0
1
.I I
-
.IO
-
C,
= .2 97 X I b q k c
1 .O2 -04
.06
.OB
.IO
.I2
.I4
.I6
-
Figure 4. Plots of l/cy us. y 1 for two concentrations of po1yst)yrenein cyclohexane. Values of c are in centimeters.
cates that the assumption of a plateau region can be considered as valid. The values of so and I?Z obtained from the leastsquares analysis of the data are given in Table 11. Those given under I were obtained when eq. 3a and b were applied, and those under I1 were obtained from application of eq. 414 T
= In
1'0
+ s o d t - n z ( ~ ~ u ~ t ) ~ (4)
Table 11: Sedimentation Coefficients and Pressure Dependence Parameter for Polystyrene 705 in Cyclohexane a t 35"
0.051 0.100
0.199 0.297
6.93 6.63 6.56 6.32
0.91
0.63 0.68 0.71
6.71 6.74 6.39 6.32
I
4
0.68 0.i9
0.46 0.71
Values of m and SO obtained by the two methods are not in very good agreement with each other, with the exception of the highest concentration. However, the correct value for so probably lies intermediate between
I
I
.6
I
.8
1.0
lo',
w2t X
,
Y-1
In
I
.2
I
1.2
I
1.4
sved-'
Figure 5. Plots of In cy2 t's. u2t for four concentrations of polystyrene in cyclohexane. Values of c are iii centimeters.
0.141
I
0.1
I
0.2
I
0.3
CONC X IO'
I
0.4
I
0.5
I
0.6
G/CC
Figure 6. Variation of the sedimentation coefficients of polystyrene in cyclohexane a t 35' with original concentration, cg. The filled circles are from schlieren data; the open circles and the triangles are the data of Table 11.
the two values as can be seen in Figure 6. This figure shows a plot of l/so us. co obtained over a tenfold range of concentration. The data represented by the filled circles were obtained under the same experimental conditions used here but with schlieren optics and calculated according to eq. 4. The values of so obtained using the interferometer are in good agreement with those obtained using the schlieren optics. Methods of improving the precision of the measurement of so are being investigated. The two major improvements that the multislit interferometer presents over the single-slit method are the increased light intensity and the ability to follow a Volume 69, Sumher 11
Sorember 1965
KOTES
4030
single fringe over the entire cell. An added feature is the simultaneous recording of both the differential and integral forms of the refractive index of the solution as a function of distance in the cell. With regard to this latter point, the optical arrangement as described here does not produce a very highquality schlieren pattern. Indeed, below a phase angle of 7.5" the resolution is not good. Preliminary investigations using a device which is composed of a horizontal slit and a series of vertical slits, side by side, as suggested by Svensson,' have shown very promising results. Other methods of improving the quality of both the interference and schlieren patterns are also being investigated. The one major drawback of the system, as compared
with the single-slit system, is that the white light fringes are no longer obtainable. In all other respects the two systems are comparable, and any inherent limitations or advantages of one apply equally to the other. The interferometric method described here permits accurate concentration measurements at concentrations that are about five times lower than those previously obtained with schlieren optics. This is of particular importance in boundary-spreading analysis where very dilute solution studies are required.lS Indeed, if longer cells were used, the lower concentration range could be extended even further. (19) R. L. Baldwin and K. E. van Holde, Fortschr. Hochpolymer. Forsch., 1, 451 (1960).
NOTES
The Enthalpy of Formation and the
Experimental Section
Dissociation Energy of Thallium Monofluoride'
The enthalpy of formation of solid T1F was determined relative to that of TI1 by measuring the enthalpy of solution of T1F in water and subsequently measuring the enthalpy of precipitation of T1I from the same solution. The literature value for the enthalpy of formation of T1I was checked in two ways: (1) by measuring the enthalpy of solution of TKOs and its Precipitation from solution as TI1 and (2) from the e.m.f. of a galvanic cell in which the cell reaction was
by Daniel Cubicciotti and Gettis L. Withers Stanford Research Institute, Menlo P a r k , California (Receieed J u n e 4 , 1966)
Barrow and co-workers2 have pointed out a discrepancy between the dissociation energy of T1F determined by them from spectroscopic data (109.5 f 0.6 kcal./niole) and the thermochemically derived value (104.1, according to data available to them). The 5.4-kcal. difference was too large to be experimental error; this implied that some of the thermochemical data were in error. In a companion paper3 we have shown that their enthalpy of sublimation of T1F was in error by 2.5 kcal./mole. (In their vaporization measurements, Barrow, et u Z . , ~ were unaware that the predominant vapor species was T12F2, not TlF.) I n order to resolve the 3-kcal. difference that still remained, we have redetermined the enthalpy of formation of T1F. With this new value and recent values for the other quantities involved, the dissociation energy from thermochemical data is found to agree with the spectroscopic value within the uncertainties in the data. T h e Journal of Physical Chemistry
Ag(s)
+ TlI(s) = AgI(s) + Tl(saturated amalgam)
and literature values for the absolute entropies of these substances. Enthalpy Measurements. A very simple calorimeter was used. It consisted of a 500-ml. dewar with a handoperated lift stirrer and a Beckmann thermometer. Two tubes closed at the bottom with rubber stoppers projected into the solution in the dewar. These contained the solids or solutions to be added, and at the (1) This work was supported by the Research Division of the U. S. Atomic Energy Commission under Contract No. AT(OP3)-106. (2) R. F. Barrow, H. F. K. Cheall, P. M. Thomas, and P. B. Zeeman, Proc. Phys. SOC.(London), 71, 128 (1958). (3) F. J. Keneshea and D. Cubicciotti, J . Phys. Chem., 69, 3910 (1965). (4) R. F. Barrow, E. A. N. S.Jeffries, and J. M.Swinstead, Trans. Faraday SOC.,51, 1650 (1955).
