403
V O L U M E 23, NO. 3, M A R C H 1 9 5 1 but with the 1 to 25 ratio, each recorded digit corresponds !o less than 0.1 mm. of travel. A simple correction factor 18 applied to the final computed slopes. Aside from reduced Cast, one striking advantage of the data printer over the recording potentiometer is the fact that one does not have t o reconvert deflections into numbers-these are printed a t exact time intervals and can bo used direotly to cornput,(!slopes in the quadratic equation. An alternative use of the data psinter has not been used in this work, hut is suited for an earlier method (5),in which the print rrlieels :ire driven continuously by a. small synchronous motor st 60 r.p.m. The telescope is not used for displacement measumment, hut. merely to view the paper, whiah in this case has a pattern of pinpoints of light projeoted on it, spaced in & square-law seequmee. Under these circumstances, the liquid boundary will cross successive target positions in equal intervals of t.ime and t.hese can he printed by the observer to the nearest 0.1 second by pressing the manual print control button. Further uses of these instruments are described in sueaeeding papers. A t this point one may well ask what justification there is for the TIE? of rdatively elaborate equipment, in studying the
motioti of liquids and solutes through paper. The authors' experience has mow than justified the time and expense involved and, indeed, indicnt,ed the utility of still more elaborate and automatic equipment. With what has heen described here, it has been possible to accumulate nn embarrassingly voluminous amount of dab&, hut in this field, the practical applications of which are almost wholly empirical md the theoretioel aspects of which have beon highly speeulnt,ive, t h w c seems to Ix nmple room fa? precise d:tt,a. LITERATURE: C l T E l l (I)Bull, H. R.. Ilahn. 71,550(1949).
J. W., and Rapt.ist, V. H.. .l. A m . Chern. Soe.,
(2) Clegg. D. L., ANAL.CHEW.,22,48 (1950). cidgg, D. L.. r h i . , 21, 192 (I!IW. (3) M U I I ~R. ~ ,H., (4) Ibid.. P. 1123. (5) Ibid., p. 1429. (6) Nestler, F. H. M.. and Cassidy, H. C;.. .1. Am. Chcrt. Soc., 72, 680 (1950). (7) Rutter, I,.. Nature. 161, .la5 (1948). (8) Williams. T. I., "Tntrodwtion to Chmmatowapby." London. Blnckie and Son.1948. R E C E ~ V Aiigiist. ED 28. 1960,
Phvsical and Geometric Factors
B.
ECAUSE paper is used in many ways and under conditions
.
.
m which its shape, texture, and the admission of liquids vary greatly, a study of some of these factors would seem to he the 6rst orderaf business in elucidating ohromat,ographlebehavior. With respect to structure, the average piece of filter paper is not merely a heterogeneous assembly of cellulose fibers, because in most cases there is a certain degree of orientation of the fibers. This arises from motion through the Faurdrinier machine, in which the suspension of fibers is felted into a cohesive mat. This preferred orientation is referred to as paper "formation," and the axis t o r a r d which the fibers tend is referred to as the machine direction and corresponds to the longitudinal travel of the main screen of the machine. Formation or preferred orientat.ion is readily demonstrated and susceptible bameasurement. t.o about l y o h y admitting a dilute dye solution t.o the paper from a very fine capillary. If the delivery is dow enough to avoid flooding, elliptical spots will he formed. For the examples shown in Figure 1, the ratio of major to minor asis is a constant and equals 1.18 * 0.014, independent of the size of the ellipse. It is also evident that the major axes 3se all inclined more or less in the same direction. For a sample of paper in which this ratio is unity-is., where circular spots are formed, there is no preferred direotion and its chromatographic behavior tiould be normal. Frequently, one encounters speeitic directions in the literature of chromatography wherein the author states that all separations are hest conducted by elution in a direction perpendicular to the machine direction. In the light of the ahove facts, this is equivalent to saying that the direction in which the rate of flow is the least (along the minor axis) is productive of good separations. Although this is a matter of empirical observation and a useful guide, it hy no means follors that a slower rate is the exclusive criterion of good separations. The upper part of Figure 1 shows how the width of the feeder strip in a Rutter disk influences t,lie mtr of flow. I n this case, two feeder strips were cut in the sZrmc: disk, one 2 mm. in width, the other 4 mm. Both feeders wore immersed in the same dye solution and removed simultaneously aftcr a suitshle interval. The two spots are elliptical in shape, again indicating appreciable paper asymmetry. The respective areas nere carefully measured
WIDTB RATIO OF FEEDER STRIPS =2:1.
\\
2mm. 4 mm.
RATIO MAJOR: MINORAXE6 = l . l R + C . C 1 4 (1.2
%I
Figure 1. Structttre of Filter Paper with a planimeter and stood m the ratio of 1.903 to 1.000. This agrees within 0.36y0with the ratio of feed strip widths. This hehavior is in complete accord with the general theory developed in these studies far the rate of flow as a function of dimensional changes. It is referrod to Inter after some additional facts are descnhed. At presont, i t seems hest to disouss the implications of the ahove behavior.
ANALYTICAL CHEMISTRY
404
Because the time of exposure was the same in both cases and the areas stand in the ratio of 2 to 1, it is evident that the respective volumes of solution in the two spots are:
Vi = raibitf V z = na2b2tj where V I = 2V2 a = major axis of ellipse b = minor axis of ellipse t = thickness of paper j = accommodation coefficient This is true becauee the area of an ellipse equals nab and f denotes a measure of the free space in the paper which can be occupied by liquid. It is now evident that in the given time interval twice as much solution must have passed through the larger feeder strip as through the smaller strip. Thus in feeding the large disk with dye solution, we must say that each strip provides an accessibility factor which is proportional to its width.
Ir
:
11u I B
G
H
I
where h = the vertical rise of liquid in millimeters (or radial spread in a Rutter disk) t = timeinminutes D = constant for a given paper and liquid b = constant and equivalent to an ho term This expression requires some explanation, in that its form arises from the fact that the square-law behavior sets in after an initial time during which the inrush of liquid is unordered and chaotic. This is no plausible assumption; it may be observed directly in the traveling microscope. Experimentally, it is also necessary to observe h and t a t some arbitrary time after the admission of liquid to the paper, after which the motion becomes uniform and follows this expression exactly, even for hours for very slow papers. Under all conditions obtaining in this work, this empirical expression is a precise measure of the results and the slope factor, D,has the dimensions of a diffusion coefficient. As will be shown later, D for a given paper can be identified directly with three properties of the liquid: its surface tension, viscosity, and density. The obvious limitation of Equation 1 is the fact that it predicts infinite height of rise in infinite time. .4n equation has been derived which is not subject to this limitation, but it is more awkward to use and, for the limited range over which these measurements extend, the two expressions are indistinguishable. This point is more fully discussed in a succeeding paper of this series. The effect of shape factors has been studied fairly exhaustively with the various shapes shown in Figure 2. From these studies, it may be said that the rate of diffusion is independent of dimension and is affected only by a change in dimension.
For example, if D is determined for the rise of liquid in a narrow rectangular strip such as A , it will be found to be independent of the width. To avoid any complications of strict reproducibility of conditions, this is best shown by using a strip of the shape shown in B. Here a rectangular strip 10 mm. wide is cut to provide two pendant strips of respective widths 3 and 5 mm. The rate of ris? in both strips is recorded simultaneously. Because identical conditions prevail, the equality of the D values may be taken ae am le proof that the rate is independent of width (Table I). Zonsider, now, the case of strip D,in which the rise occurs initially in a width of 10 mm. and then abruptly enters a region of half this width ( 5 mm.). After an intermediate zone in which the liquid is perceptible syirling about, the rate settles down to a new value which is higher than before. Plotted results are shown in Figure 3.
J
-
Figure 2. Shapes Used to Study Effect of Geometric Factors on Rate of Diffusion of Liquids through Paper
EIRECTION O F FLOW
A more general treatment shows that the effect of geometric factors may be explained completely and quantitatively by assuming that the rate of flow is controlled by two factors: (1) accessibility, as controlled by the dimensions of the feeder, and (2) capillarity, as controlled by the dimensions of the succeeding portions of the system. This is investigated more exhaustively in the geometric patterns illustrated in Figure 2. Before any quantitative study of geometric or shape factors could be made, it was necessary to establish some relationship between motion of the liquid through the paper and time. Empirically, one may use the fact that the square of the distance (height of rise, or radial spreading) is directly proportional to time ( 8 ) . This relationship is accurate and reproducible and provides rates well within the experimental precision which, in itself, is high. Actually, this empirical rule represents a special limiting case of the general law governing flow through the paper, and those relationships are discussed in a subsequent paper in this series.
5r
J
4
I
I
I
I
8
12
16
20
;II\!E
With more than adequate precision we may write:
ha = Dt
-b
Figure 3.
(1)
IN MINUTES
Gain in Diffusion Rate for Strip with Abrupt Dimensional Change
V O L U M E 23, N O . 3, M A R C H 1 9 5 1 When R >>1, this reduces to: Table
I. Effect of Width of Strip
8. & 8. paper 598 YD 3- and 5-mm. rectangular strips, Type B , Figure 2 Temperature 25' C. Eluant. Ethvl alcohol-water 84.6% n s 5 = 1.3630 Time, JIin.
Net Rise, hIm.
1
a
7 9
11
13 15 17
Time, Net Rise, Min. Mm. 0 0.0 2 18.9 4 31.5 40.3 6 47.4 8 53.4 10 58.7 12 63.9 14 68.4 16 72.0 18 K i d t h = 3 mm. D = 314 mm.Z/min.
hz 96 681 1297 1945 2550 3192 3832 4476 5098
26.1 36.0 44.1 50.6 56.5 61.9 66.9 71.4
Width = 5 mm. D = 315 mrn.2,'min.
Table 11.
Effect of Shape of Strip
Time, Min.
h'et Rise, Mm.
0 1
2 3 4 5 6 7 8 9 10
11
12 13 14 15 16 17 18 19
1
When R, = 10, -= +
h2 000 357 992 1624 2247 2852 3446 4083 4679 5184
h2
0 0
0.0
3.2 12.0 18.5 23.7 28.2 31.9 35.6 38.8 42.0 45.0 47.6 50.3 53.0 55.8 59.0 61.7 64.3 67.6 70.2 DI = 246 mm.z/min. D Z = 351 mm.Z/min.
10.24 144.0 342.3 561.7 795.2 1017.6 1267.4 1505.4 1764.0 2026.0 2265.8 2530.1 2809,O 3113.6 3481.0 3806.9 4134.5 4569.8 4928.0
In this and all other variations, it may be shown that the gain, G, in diffusion rate is predicted by
G = -1
+ R2 2R
where R is the ratio of the greater to the smaller width.
R
=
WW/WN
For decreased rate-that is, flow from a narrow to a wider zonethe new rate is the reciprocal of this 1/G. This simple relationship is derivable from the following considerations.
If it is assumed that the rate of flow is the sum of two factors, the first of which we may call the capillarity factor and the second the accessibility factor, then in the case of the transition from a wide to a narrow strip, the capillary factor will be l / R and the accessibility factor will be R. Hence, the new rate will be:
V2
=
(k
+R)
The original rate will be governed by conditions wherein both of these factors are unity. I n other words,
W w
=
W N or R = 1
Hence VI
= 1
+1 =2
Defining the gain, G,as G = Vz/V1,we have:
G = -1
+
R2
2R
R2
2R
R 5.05 and - = 5.00 2
Hence, for width ratios of 10 or greater, the use of Equation 3 instead of 2 leads to an error from this approximation of 1% or less. The calculated and observed rates for the example plotted in Figure 3 are included in the diagram, and the data from which these rates are computed are shown in Table 11. Returning to the simple case represented by A , the question may be raised about the uniformity with which the strip is cut. IIost chromatographers stress the fact that their sheets or strips of paper have been machine-cut. This is undoubtedly more convenient, but how does it affect the precision with which rates may be measured? This work indicates throughout that changes in dimensions are important. Measurements were, therefore, made on strips of the same dimensions of form A , some cut by machine and others with a pair of scissors. In the latter, no particular care was taken to preserve the maximum uniformity other than identical average dimensions. Typical results are shown in Tables I11 and IV.
-
Diffusion Coefficients Machine-cut strip D 203.5 mm.*/min. Hand-cut strip D = 203.8 mm.*/min.
*
Table 111. Effect of Uniformity of Strip 8. & S. 598 ST Machine-cut 10-mm. rectangular strip Eluant. Ethyl alcohol-water 84.6% nL6 = 1.3630 T = 25' C. Time, Min.
5 6
7
5
9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27
?jet Rise, Mm.
24.9 28.5 31.4 34.0 36.6 38.9 41.3 43.7 45.9 47.9 50.2 52.1 54.1 55.9 57.9 59.8 61.6 63.2 64.9 66.5 67.8 69.5 70.8 D = 203.5 mm.l/min,
620 812 986 1156 1340 1513 1706 1910 2107 2294 2520 2714 2927 3125 3352 3576 3795 3994 4212 4422 4597 4830 5013
This unusually close agreement indicates that there is no detectable difference. Indeed, from what is known about the influence of dimensional changes, one could predict with reasonable certainty that a strip cut on both vertical edges with a seamstress' pinking shears and thus having uniform serrations would yield rates identical with a plain rectangle. Actually, a t each serration the rate would rise, only to fall a t the next, and so oh. There is a slight indication in the data of Tables I11 and IV that the average step-by-step variation in slope in the hand-cut strips is about 1.24 times greater than in the machine-cut strips, although the average slopes are identical to within 3 parts in 2000. This is p r s
ANALYTICAL CHEMISTRY
406 -
Table I \ . .
Effect of IJtiiforiiiit>- of Strip
S. Jr 9. 398 ST Hand-cut 10-rnIn. rectangnlar strip Elirant. Ethyl alcohol-water S4.CiL'; Teinperatirre 2 5 O C. = 1.3630
::n
S e t Rise.
Tllne.
Min.
IIm.
0
0.0 8.5 15.3 20.5 25.3 29.2 32.8 36.0 39.1 41.9 44 6 46.7 49.0 51.4 53.1 55.1 56.9 58.5 60.3 61.8 63.2 64.5 65.8 67.8 69.2 71.0 72.4
1
2 3
! 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
--23
11 9.>
24 25
26
It
0 72 234 420 610 8.53 107ti 12Yti l52Y 1756 198Y 2181 2401 2641 2820 303ti 3238 3422 3636 3820 3994 4160 4330 4597 478Y 504 1 ,5242
T h r ~( w e repivsc~iitrdlry .I, Figurt. 2, in w1iic.h a rectaiigu1:ir strip of 10-mni. width gratiuall~,tapers to a Bmm. width and t l w i i once inore tapers 11ac.kto thr origind width, I)ehaves in a i)rcdic~twl)l(~ niaIiner. Typicaal dnt:i :ti'i- g i v w i i i Tahlt. \., A group of three str:iiglit liner results froni thca syuare-hw plot. \\'lien the liquid riaea beyond the r.ect:ingle into t h r taper, it s1)reds up. [.pori leaving the constriction, thr. r:itia i 1 e C i w w w . ' l ' t i t s d:ct:I >,itJld :I w l u e for the slower rate of:
D
= 206 iiini.2,iniin
:i11(1 for {hi. r:itr i)rwrding it oi:
D = 237 nini.Viniii. (;:iiii Gi. t l i r i ~ t ~ ~ o r v : GuirSd.
I
I >
l i t , c*:iliwl:ittvl
= 0.8i
g:iiii, accwrding to Equation 2, is:
GcniC,i.= 0.58
In another example for the sanie shape, J , the d:lta nlav cutmined in another way. The data of Table V I yield { h r w .tr:iiglit linw, the respective valuep of D of which lire: l j t b
13, = 209 nini.?/niin. 112 = 226 niin.2,fmin. 11:; = I94 nini.*/min. A s bdore, it the gain, as defined :Lnd calculuted i i i Lyuation 2, i* wed to predict the slowest w t c i n ternis of' the rate immedi:itc.ly puvedirig i t , then:
Go,,3
