Determination of copolymer composition by a polarization diffusometer using monochromatic light of different wavelengths

DETERMINATION OF COPOLYMER COMPOSITION BY A POLARIZATION DIFFUSOMETER USING MONOCHROMATIC LIGHT OF DIFFERENT WAVELENGTHS* G. A. FoMII~ A. A. Zhdanov State University, LeningrsAi

Received 13 April 1972) A method is proposed for determining the composition of copolymers, which is based on the additive nature of refractive index increments of copolymer components. According to this method, in order to increase the sensitivity and accuracy of determining composition, the increments are measured using a Tsvetkov polarization diffuse. meter in monochromatic light of different wavelengths. W~EN examining the physical behaviour of copolymers one of the i m p o r t a n t problems is the determination of composition. To determine the composition of copolymers, the refractive index increment v of the copolymer studied a n d increments vl and v~ of components 1 a n d 2 are often measured b y a refractometer a n d composition is calculated assuming t h a t increments vx and vz [1, 2] expressed b y the ratio

v=xvl q- (1-- x)v2,

(1)

are additive, where x is the gravimetric proportion of component 1. The rofractomotrie method, however, m a y involve considerable errors in determining composition if the oopolymer contains even small proportions of impurities, which m a r k e d l y differ from the copolymor in refractive index and therefore considerably alter the overall increment [3]. The accuracy of determining copolymer composition is considerably increased and the effect of i m p u r i t y on t h e overall increment m a y be considered on u s i n g interference t y p e diffusometers [4] with a Guy interforometer a n d a Tsvetkov polarization diffusomoter [6]. This paper examines possibilities of further increasing the accuracy of these measurements b y using monochromatic light of different wavelengths. The applicability of the methods proposed is illustrated b y measurements of composition of numerous fractions obtained from three graft copolymer specimens of methylmethaery]ate and styrene (MM2k-

St). Determining the refractive index increment by a polarization diffusometer and considering the effect, impurit/y. Polarization d i t ~ s o m e t e r s are used for accurately determining the difference in refractive indices of solution n a n d solvent n, to very low values of A n o n - - n , -----(0.5--1-0) × 10 -5. The system of interference curves J(~) plotted b y using a diffusomoter is related to the distribution of gradient of the refractive index dn/dv in the height of the cell t [6]. I t follows from the t h e o r y of polarization diffusometers t h a t area Q enclosed under the interference curve is constant for a monodispersed specimen during the entire experiment a n d is Jnhab q=---y--, (2) * Vysokomol. soyod. A15: No. 8, 1917-1925, 1973.

2165

2166

G.A.

FOMIN

where h is the length of the diffusion cell along the course of the ray, a - - t w i n n i n g of spar, b - - d i s t a n c e b e t w e e n a d j a c e n t interference bands of t h e B a b i n c o m p e n s a t o r and 2 - - t h e w a v e l e n g t h of the light used.* The direction in which the interference bands are d i s t o r t e d depends on the sign of t h e increment, i.e. it is t h e reverse w h e n no >n8 a n d w h e n no
-

c

habc

(3)

The sensitivity of this m e t h o d of d e t e r m i n i n g composition m a y be c h a r a c t e r i z e d b y the value d_Q_Q:Q ( v l - v,) dx

(4)

~,

I t can be seen from e q u a t i o n (4) t h a t sensitivity increases w i t h a r e d u c t i o n of v of t h e cop o l y m e r in t h e solvent used if t h e high value of Q is m a i n t a i n e d b y increasing solution c o n c e n t r a t i o n (and using a c o m p e n s a t o r w i t h a fairly high b value). I f t h e v a l u e of v is n e a r l y zero, a different i n c r e m e n t sign, i.e. a v a r i a t i o n in the direction of t h e interference c u r v e to t h e opposite, corresponds e v e n to a small d e v i a t i o n of the composition of a c o p o l y m e r fraction in one or a n o t h e r direction, according to zero increment. I f t h e dissolved substance contains m c o m p o n e n t s w i t h different coefficients of diffusion D~, r e f r a c t i v e indices nob and c o n c e n t r a t i o n s c~, the interference c u r v e is t h e superimposition of curves corresponding to individual c o m p o n e n t s and in t h e initial m o m e n t t h e area enclosed corresponds to t h e t o t a l i n c r e m e n t of t h e solution. hab m m Q = - - ~ An~= ~-~ Q~ 2 i-1 i-1

(5)

W i t h fairly different D~ values the low molecular components, which cause t h e format i o n of " w i n g s " at t h e interference curve, as t i m e goes on, are regularly d i s t r i b u t e d along t h e h e i g h t of t h e diffusion cell a n d t h e corresponding c o m p o n e n t s Q~ b e c o m e zero, while the r e m a i n i n g area only depends on high molecular w e i g h t components. Consequently, Q, values can be d e t e r m i n e d (e.g. b y p l o t t i n g the relation b e t w e e n Q and t h e t i m e of diffusion t) and f r o m e q u a t i o n (2)--An~ of each c o m p o n e n t established. I f t h e r e f r a c t i v e i n d e x no~ a n d d e n s i t y p~ are known, c o n c e n t r a t i o n c~ can be d e t e r m i n e d since nob--ns z i n ~ c~

(6)

P~ I f a low molecular weight i m p u r i t y is present in the copolymer, this m e a n s t h a t : 1) t h e e x p e r i m e n t should be carried out for a fairly long t i m e in order to blur t h e wings of t h e interference c u r v e caused b y t h e i m p u r i t y ; 2) w h e n calculating composition it is necessary to use areas Qp d e p e n d e n t on t h e p o l y m e r and o b t a i n e d after t h e blurring of wings, which (in a d d i t i o n to visual evaluation) is controlled according to t h e c o n s t a n c y of area during the subsequent course of t h e e x p e r i m e n t ; 3) strictly speaking, w i t h this calculation p o l y m e r concentration Cp=C--Ctmp, (7). * I n t h e f o r m given formula (2) is valid w i t h a magnification factor of the a p p a r a t u s y ~ 1, which was m a i n t a i n o d in the study.

Determination of copolymer composition

2167

and not the overall solution concentration e should be used, where eimp is the concentration of the low molecular weight impurity, which can be determined from equation (6) if the refractive index romp is found by independent experiments [5]. F o r example, a solvent can be selected in which the impurity has a zero increment and cannot therefore be seen on the interference curves. Subsequent results show t h a t for the fractions studied em~ hardly altered the result of determining composition.

FIG. 1. Interference curves plotted during the diffusion of fraction 4 of specimen 0 in bromoform: (c=0.736 x 10 -~ g/cm3): a - - i n the white light of a DRSh-250 mercury-arc lamp ( t = 5 2 min); b - - i n white light of an incandescent lamp ( t ~ 5 3 min). Here and in Figs. 4-7 the upward direction corresponds to the positive sign of the increment.

Use of monochromatic light of different wavelengths in methods requiring a polarization diffusometer. A shortcoming of the method examined using " w h i t e " light is the fact t h a t th8 variation of the refractive index increment is not taken into account, which has a particularly marked effect near a copolymer composition, which corresponds to zero increment in a given solvent. As a result, the point of the interference curve of zero order appears to b0

............

~

\.. ...........

/ -

-

°

°

÷[

.\ °"

0

/ ... \ ............

•

o. . . . .

~ . . ' ~ , ~"f- f

FrG. 2. Layout explaining the formation of a sharp m a x i m u m of the interference curve in white light, the order of which is different from zero. For clarity the white light consists o f two components with a shorter (~1) and longer (2,) wavelength; corresponding interference curves are shown by broken lines (~1), lines of dots (~,) and places of superposition by a continuons line; the figures on the right hand side are the numbers of interference order. blurred (Fig. la) which reduces the accuracy of measuring Q. The top of the interference curve m a y be well defined in an order other than zero which explains Fig. 2, b u t the "base line" itself appears to be more or less blurred since the compensator has some variation for orders other t h a n zero. Furthermore, on using various sources of non-monochromatic white light, Q values are different (Fig. l a and b) which is due to a different energy distribution of radiation in the spectrum. N o t only the refractive index increment v of the MMA-St graft copolymer, b u t also t h e increment v l o f polystyrene (PS) in bromoform shows a significant variation. Therefore as well as making measurements in white light for m a n y fractions, particulaxly near the zero increment, measurements were also made in this study in monochromatic

2168

G . A . FOMI~

light of different wavelengths. Therefore, in the optical system of the polarization diffusometer, in addition to the conventional white light s o u r c e - - a n incandescent l a m p - - a DRSh-250 mercury lamp was also used, the radiation of which if necessary was directed to a condenser b y means of a rotary prism. The lines of the mercury spectrum were separated by corresponding pairs of light filters made of coloured glass (4 = 578 n m - a combination of light filters ZC-7 and OS-13; ~ 5 4 6 n m - - O S - 1 1 and PS-7; ), 4 3 6 n m - - Z h S - 1 1 and SS-4; 2 = 4 0 5 n m - - P S - 9 and BS-8). The apparatus was very powerful and the degree of monochromatization sufficient for the purpose: the m a x i m u m of the interference curve became well defined. TABLE 1.

R E F R A C T I V E I N D E X I N C R E M E N T S OF P S A N D

P P M A IN BROMOFORI~I

,~X 10 -5

vl, cm3/g

r2, cma/g

~ X 10 -5

~h, cm3/g

v2, cnP/g

578 546 436

0.0111+0.0002 0.0115_+ 0.0002 0.0142_+0.0002

--0.0812 -- 0.0812 --0.0940

405

0.0153+0.0003

--0.0983

White light

0.0123

--0-0866

Three MMA-St graft copolymer specimens conventionally termed 0 [7], A and B [8], which were prepared by methods previously described [9] and have a comb structure, were examined. Since the graft eopolymer fractions prepared contained an overwhelming proportion of styrene (x ~0-9), bromoform was used as solvent when determining composition, which ensured a low value of v and according to equation (4), an increased accuracy in determining composition.

- ~/0fcm~g a 1"5

b

H-

/

/

1.a /

1.1It

/

I

5

7 ;

~

/

I 5

l B .~ . [O6cm

FIG. 3. Relation between the refractive index increment of polystyrene (a) and polymethylmethacrylate (b) and the wavelength.

Determination of refractive index increments for polystyrene and polymethylmethaerylate using monochromatic light of four different wavelengths. I n separate experiments at 21 ° first the refractive index increments of PS and polymethylmethacrylate (,PM.MA) were determined in bromoform, and denoted as vl and va, respectively for monochromatic light of four different wavelengths and for the white light of an incandescent lamp. Bromoform was first purified by v a c u u m distillation; for all the experiments it was taken from the same batch. To determine v2, PMMA was used which was the basis of graft copolymer specimea

Determination of copolymer composition

2169

0 a n d to determine vl, four experiments were made with two different high molecular weight PS fractions (M[~]=4× 105 and 1 × 106), the results of which are averaged. Calculated values of vl and vz are shown in Table 1; for vl, according to results of four experiments errors were given which confirm a satisfactory reproducibility of results. I n the graft copolymer specimens studied lateral PS is of low molecular weight a n d taken separately, has a lower increment [10] t h a n the high molecular weight PS. However, according to results formerly obtained [5], vl in the graft copolymer has a value which corresponds to high molecular weight PS. The relation between values v~ and v~ and wavelength 2 is shown in Fig. 3 and takes the form which is normal for the variation of refractive index increment. Vv-hen increment is calculated by equation (3) for white light there is some u n c e r t a i n t y in the equivalent value of 2 due to possible variations in energy distribution of the white light source in the spectrum and spectroscopic sensitivity of photographic plates used. Since equation (3) can be written in the form

v=k~,

(8)

k=Q/habc,

(9)

where the value is proportional to the increment, the equimolecular value of 2 for white light is determined from experimental values k ~ = 0 . 2 5 1 × 1 0 a and k2=1"77×10 s cm2/g by plotting a linear relation k2(2) shown by a broken line in Fig. 3. Points of intersection with the experimental curve of increment variation v(2) in both cases are close to 2 = 4 . 9 ~ 1 0 -s era. However, in the final calculation of composition b y the formula which follows from equation (1) Y-- Yl

x=--

(10a)

VI--YS

the value of 2 is reduced and calculation can be made by the formula

k--ks x= --

kl--k

(10b)

Determining the composition of fractions. Solution concentrations of graft copolymer fractions of MMA-St in bromoform were (0.7-1-6) × 10 -2 g/era s, which ensured the requisite accuracy of determining the areas under the interference curves. Experiments were carried out at 21 o. For all fractions of specimen 0 and some fractions of specimens A and B showing a marked variation, measurements were made not only in the white light of the incandescent lamp, b u t also in monochromatic light of four wavelength. The experiments lasted for 7 to 20 hr, which was quite sufficient not only for blurring the "wings" of interference curves, b u t also for controlling the constancy of the area during the subsequent experiment. To determine the refractive index of a low molecular weight impurity nimp, experiments were first made with fraction 3 of specimen A in two solvents--bromoform and bromobenzene. Solving the equation system Qlmp(£=(nimp__ns,) c J ( i = 1 , 2) hab pimp

(11)

(f=Clmp/C is the gravimetric proportion of the impurity) enables us, in principle, to calculate n~mp. However, results of these experiments cannot agree with each other, considering t h a t f = const. Control experiments with the same fraction in bromoform confirmed t h a t the ratio Qlmp/C does not remain constant, i.e. in different successive samples of the same fraction the value o f f somewhat varies.

2170

G.A.

FoMI~

The following measures were therefore used to find n~,p: for each of the three copolymer specimens a solvent was selected, in which the "wings" of the interference curve correspond° ing to the impurity disappeared. For all 0, A, B specimens the wings of the curve disap-

4

~ ~ i ~

~'~i~iii~!!i ¸i~!i

FxG. 4. Interference curves plotted during diffusion in bromoform of several fractions containlng a low molecular weight impurity with v~np< 0 and different t values: A - - f r a c t i o n 1 of specimen B (e=0.895x 10 -2 g/cm 2) with v < 0 ( 1 - - t = 4 5 rain; 2-3.5 hr; 3-13 hr, during the experiment the compensator was changed); B - - f r a c t i o n 2 of specimen B (c=0-900 × 10 -~ g/cm 8) with v > 0 (1--1 hr 45 min; 2--15 hr 15 rain). peared i n benzene, with a corresponding value of nimp= 1.5. This is in agreement with the fact t h a t grafting was carried out in toluene and fractionation in benzene (specimen 0) or in a benzene-chlorobenzene mixtures (specimens A and B) and the copolymer m a y have contained small amounts of absorbed solvent with n,---- 1.5. The value of nimp being known and Q~np being measured on interference curves, the value of Campwas found from equations (2) and (6) and cp from equation (7). Figure 4 shows

FIO. 5. Interference curves plotted during diffusion in bromoform of fraction 3 of specimen B: a - - a f t e r drying from a solution in benzene (e-----0.755× 10 -2 g/era s, 1 hr 5 rain); b--after lyophilic drying from a frozen solution in benzene (c = 1.20 X 10-' g/em s, 1 hr).

Determination of copoIymer composition

2171

as a n example the interference curves for some fractions containing an impurity, the increme n t of which in bromoform is negative, the curves being V-shaped with a eopolymer increment of v < 0 and W-shaped with v > 0; during the experiment the wings of these curves became blurred. Corresponding gravimetric proportions of the impurity for fractions of specimens A and B, for example, did not exceed 6 a n d 12yo respectively. Since the value of nimp -~ 1"5 was found approximately a n d the value of p~np~0"88 g/cm 8 was taken assuming that benzene was the impurity, the error in determining f values was ~ 40 ~o. However, this has practically no effect on the calculation of composition x, because correction Ax itself, taking into account the concentration reduction of the polymer owing to the presence of impurity, is Y

~x=

f, i} 1 - -

(12)

Vs

which only approximates to 0.005 for ~ractions 2 and 3 of specimen B, while in the remaining eases, is noticeably lower. As can be seen from equation [12], this is due to the low value of v of the copolymer in bromoform.

A

2

FIG. 6. Interference curves plotted in monochromatic light of different wavelengths for fractions 4 (c~0"763 × 10 -8 g/cm s) (A) and 6 (c----0.92 × 10 -8 g/cm s) (B) of specimen 0 in bromoform with 2 × 10~=578 (•); 546 (2); 436 (3) and 405 n m (4): A: t-- 1 hr 26 m i n (•); 1 hr 31 m i n (2); 1 hr 23 m i n (3); 1 hr 24 min (4). B: t - - 5 hr 29 rain (•); 5 hr 25 rain (2); 5 hr 27 rain (3); a n d 5 hr 28 rain (4); fraction 6 contains a low molecular weight impurity which is noticeable from the bend of lines.

2172

G.A.

FOMIN

I t should be noted that the impurity Content is m i n i m u m (and is often zero within the limits of experimental accuracy) in those fractions which were dried lyophilicaUy under vacuum from a frozen solution in benzene [11]. A control experiment with fraction 3 of specimen B indicates that after lyophilic drying impurity content decreased approximately 4.5 times (Figure 5).

FIG. 7. Interference curves plotted during diffusion of fraction 3 of specimen A in bromoform at 24 ° and c=1.1 × 10 -~ g/cm3: a - - A > 6 0 0 n m (KS-10 light filter), v<0; t-----2hr 49 rain; b - - 2 = 4 0 5 nm, v > 0 ; t = 4 hr 9 rain. Subsequently, the compositions of fractions were determined using Qp and ca according to equations (3) and (10a). Figure 6 shows typical interference curves plotted in light of different wavelengths for two fractions near the zero increment. As a result the variations TABLE

2.

REFRACTIVE

INDEX

INCREMENTS,

COMPOSITION

AND

F R A C T I O N S OF S P E C I M E N S Q , A A N D

f~, Fraction, NO.

578

5 4 6

v x 10'. cmS/g

x

1'46 0.92 0"56 0-25 - 1'37 - 4'28

0.896 0.890 0.886 0.882 0.865 0.833

~x-~. i emS/g I

MOLECULAR

WEIGHTS

lira 4 3 6 [

x

, x liP. cmS/g

0.893 0.887 0-883 0.879 0"865 0.832

2"70 2'17 1'90 1'35 -- 0"37 -- 3"93

~

405 v X 10'. ema/g

[

whitellgh_t

j~

Msa×lO_ .

x

v x 10*. cm*/g

x

0"894 0"889 0'888 0.883 0.865 0"832

2"11 1-76 1-18 0"74 -- 0"69 -- 4'07

0-897 0"893 0'888 0'883 0"869 0'834

0'895 0.890 0'886 0'882 0.866 0"833

697 606 471 351 175 80

--- 1"17 -- 0"78 0"54 1'18 2.38 2.63 3.52 4-28 4'26

-0"863 0"869 0"882 0-888 0.900 0.902 0.913 I 0.919 I 0-919

0'849 0"863 0'859 0.880 0.887 0.900 0.902 0.013 0-919 0.919

1960 1430 775 321 168 83'5 57'5 47 "7 37"7 26"0

--1.36 4.05 4.65 4"39 4.94 5.08 5-16

0"862 0'917 0.923 0.921 0.926 0"928 0.928

0'862 I 0.916 / 0-922

1310 390 228

0-926 / 0.928 / 0.928 [

107 76'0 21'0

Sample 0 1 2 3 4 6 7

li /

3 4 5 6 7 8 9 10

0 0.67

0.880 0.887

1-57 1 '03 0'68 0'33 -- 0"96 --4'15

0"893 0"889 0"886 0-881 0"865 0-832

3.42 2'67 2'55 2'05 -- 3'82

Sample A --

0

0.876

0'76

0.884

1"21 2'09

0"880 0'888

1"86 2'75

0.882 0.889

5-84 6.44

-0.917 0.022

Sample B 2 3

3"29 3.86

4 5 6

--

0.915 0'922

3'47 4'16

0"913 0.921

5"20 5"79

0"917 0'921

---

--

OF

B

[

I

t

* The composition of fraction 1 of specimen Ai was d e t e r m i n e d f r o m analytical results for m e t h o x y l groups.

D e t e r m i n a t i o n of c o p o l y m e r composition

2173

of r e f r a c t i v e i n d e x i n c r e m e n t of the area of p o l y m e r Qp differ m a r k e d l y for the same fraction, b e c o m i n g zero in some cases (Fig. 6, B4). F i g u r e 7 illustrates a case w h e n a fraction of certain composition shows different signs of i n c r e m e n t at different wavelength. v values at different w a v e l e n g t h s and in white light of an incandescent l a m p a n d corresp o n d i n g x values are shown in Table 2 for t h e fractions of three M M A - S t graft c o p o l y m e r specimens studied: the s a m e Table indicates a v e r a g e results of all m e a s u r e m e n t s of ~c values. F o r those fractions, of which the r e f r a c t i v e index i n c r e m e n t was only d e t e r m i n e d in wite light, this c o l u m n gives t h e compositions found from the i n c r e m e n t in white light. The v a r i a t i o n of limiting values of x from ~ was under ~:0.003. Since x values d e r i v e d for t h e s a m e fraction from m e a s u r e m e n t s at different wavelengths, differ from each o t h e r in the t h i r d sign after t h e c o m m a , the error in finding correction zfx, which is due to t h e i n a c c u r a t e v a l u e of nlmp a n d pimp, is insignificant. A t t h e same t i m e as testing the effect of lyophitic d r y i n g on i m p u r i t y c o n t e n t in two e x p e r i m e n t s w i t h fraction 3 of specimen B, t h e reproducibility of results of d e t e r m i n i n g composition was elucidated (in the first e x p e r i m e n t m e a s u r e m e n t s were only m a d e in white light, while in the s e c o n d - - a l s o in m o n o c h r o m a t i c light of four wavelengths). The x3 values d e r i v e d differed by 0-002 (Table 2). Dispersion curves of the i n c r e m e n t of fractions are shown in Fig. 8.

w/O 3,cm~/g

cl

b

2t?

os

-5 FIG. 8. R e l a t i o n b e t w e e n the r e f r a c t i v e index i n c r e m e n t of fractions and wavelength: a - - f o r fractions of specimen 0; b - - f o r several fractions of specimens A and B. The figures at the curves d e n o t e the n u m b e r of fraction, t h e letters denote the specimen. The m e a s u r e m e n t of refracti~ e i n d e x increments at different w a v e l e n g t h s reduces t h e error in d e t e r m i n i n g composition, since the areas of interference curves p h o t o g r a p h e d in m o n o c h r o m a t i c light can be m e a s u r e d m o r e accurately; a change in t h e direction of interference curves to t h e opposite for different w a v e l e n g t h s takes place w i t h different compositions; n o t one, b u t several e x p e r i m e n t a l values of x are o b t a i n e d in each e x p e r i m e n t , which are t h e n averaged. The last c o l u m n of Table 2 shows molecular weights MaD of fractions studied, which were d e t e r m i n e d [7, 8] from diffusion and s e d i m e n t a t i o n d a t a of fractions in b u t y l acetate. T h e c o m p o s i t i o n of fractions of all specimens varies s o m e w h a t w i t h a change in molecular weight; consequently, initial g r a f t c o p o l y m e r specimens were heterogeneous in composition, b u t t h e h e t e r o g e n e i t y was slight. Diffusion analysis enabled us n o t only to d e t e r m i n e t h e composition of c o p o l y m e r fractions, b u t also to e v a l u a t e composition h e t e r o g e n e i t y [5, 12]. I f composition h e t e r o g e n e i t y is due to h e t e r o g e n e i t y in diffusion coefficients (molecular weights), it is n a t u r a l to e x p e c t in the case of a g r a f t copolymer, the molecular weight of which M=MJ(1 --x) (M2 is the m e -

2174

G.A.

FoMI~

lecular weight of the base), t h a t during diffusion with a slight increment the diffusion curve $(~) "is split": it acquires a typical W shape and when v l > v l , the p a r t slowly becoming blurred (central p e a k with a positive increment) is due to the high molecular weight component of the copolymer examined a n d the p a r t of the curve which becomes blurred a t a higher rate (wings with a negative increment) is due to the low molecular weight component of the copolymer. Maximum possible separation of the curve J (~) (equality of areas of parts of t h e curve with positive and negative increment) during diffusion of a eopolymer heterogeneous in composition, should take place in a solvent, where the refractive index increment of the copolymer v = 0 [12]. Consequently, the selection of a solvent for the most accurate analysis of composition heterogeneity coincides with the most accurate determination of composition, which is expressed b y equality (4). The use of monochromatic light of various wavelengths helps in approximating to zero increment. F o r the graft copolymer fractions of M M A - S t studied during diffusion the curve J(~) did not separate into parts with different signs of increment even when v = 0 (e.g. Fig. 6 Bg). This means t h a t the fractions are not noticeably heterogeneous in composition due to hetero. geneity in molecular weights. W e note t h a t the separation of the diffusion curve for a copolymer of heterogeneous composition should n o t be confused with an outwardly similar effect o f . i m p u r i t y "wings", which when vp > 0 and v~apd 0 also make tl~e curve $ (~) W-shaped (Fig. 4 / 3 1). Since the diffusion coefficient D~np >>Dp, the i m p u r i t y wings of the curve J (~) are clearly seen practically immediately after the beginning of the experiment and disappear after a few hours (Fig. 4 B2; Fig. 6 B4), while the separation of the curve ~(~) in the case of composition heterogeneity only begins to be apparent during this time. Finally, we wish to t h a n k V. N. Tsvetkov for the proposed theme of the study.

Translated by E. SE~.ERE

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