Boundary extension during sedimentation of polymer solutions

METHODS OF INVESTIGATION B O U N D A R Y EXTENSION DURING SEDIMENTATION OF POLYMER SOLUTIONS * I. K •EKRASOV All-Union Scientific Research I n s t i t u t e for Synthetic Fibres

(Received 19 October 1970) RAPID sedimentation is one of the most important methods for determining molecular weight dastnbutlon (MWD) m polymers [1, 2]. Practical methods now used for calculating MWD from sedimentation curves are considerably m advance of theory. This is first of all due to the difficulty m developing the theory of extended boundaries of polydispersed polymer solutions for terminal concentrations. The "non-Ideal" state of the system a n d (or) the dependence of 8 = s (c) (s is the sedimentation coefficient, e, concentration) considerably complicate analysm, therefore, theoretical treatment [3-5] has been gaven to the sedimentation of maeromolecules m a 0 solvent without the concentration dependence of sedimentation coefflclents. On the other hand, for several reasons, the selection of 0 solvents for sedimentation is an extremely difficult, sometimes insoluble problem and sedimentation experiments should be carrmd out with polymer solutions m suitable solvents I t is i m p o r t a n t m this case to elucidate the causes of the extension or retraction of the b o u n d a r y , which enables the procedure for calculating MWD to be selected. As a first step b o u n d a r y extension should be studied using polymer fractions during the sedimentary e x p e r i m e n t [6, 7]. Boundary extension during sedimentation of polymer solutions. Using poly-m-phenylenelsophthalamlde (PPP) fractions m dlmethylformamlde (DMFA), dimethylacetamlde (DMAA) a n d m DMFA ~ 0.25 ~ LICI and for polyacrylonltrlle (PAN) fractions m DMFA, a s t u d y was made of the dependence of ~s/2t (~2 m the standarddevlation) on rat (r is the average coordinate of the gradient curve, t--time). Sedimentation experiments were carried out in a MOM G-100 ultracentrifuge with a phase-contrast plate at 50,000 rev/mm. Concentration varied between 0-007 and 0.008 g/dl for P P P and the concentration of P A N varied from 0 10-5 g/dl. The value of ~2 was calculated for each photograph using " P r o m m " ETsVM from 13-16 points according to the formula

~= f

(r--~)2q(r) (r/ro)~dr,

(1)

where r0 is the coordinate of the menlseus, factor (rite) 2 takes Into account sectoral dilution a n d t h e standaxdlzqd distribution function of displacement q(r) is determined by equation ~e 1 {¢9n\ q(r) = ~ r = ~ n ~ - r ) '

(2)

where e and n are the concentration and refractive Index of the polymer solution, respectively, m the cell at chstanee r from centre of rotation and An is the &florence between the refractive radices of solution and solvent. *Vysokomol. soyed. A 1 4 : N o . 10, 2252-2258, 1972 2636

B o u n d a r y e x t e n s m n d u r i n g s e d i m e n t a t i o n of p o l y m e r solutions

2637

T y p m a l relations between ~]2~ and r25 are s h o w n in Fig. la. Since P P P solutions were centrifuged for 40-45 m m a n d t h e first p h o t o g r a p h was t a k e n a few m i n u t e s a f t e r r e a c h i n g 50,000 r e v ] m m a n d t h e dlstfllatmn t u n e was 15 ram, it is necessary to t a k e into a c c o u n t t h e correct dlStfllatmn t~ne. T h e simplest m e t h o d is to a d d 1/3 distillation t i m e to t h e t i m e of centrifuging c o u n t e d f r o m t h e m o m e n t of r e a c h i n g t h e r e q m s l t e r a t e of r o t a t m n of t h e rotor. A c c o r d i n g to t h e second m e t h o d t h e correction for dlstlllatmn is f o u n d f r o m t h e dependence of In rm on t(r~ is t h e m a x i m u m coordinate of t h e g r a d m n t curve) [1]. F m a U y , according to t h e t h i r d m e t h o d [8] t h e F u j l t a e q u a t i o n [9] is applied, according to w h m h t h e effect of h y d r o s t a t m c o m p r e s s i o n on s e d u n e n t a t l o n coefficients m t a k e n into account. The d e p e n d e n c e of In (r/ro)]o~5(o9 is t h e a n g u l a r v e l o c i t y of t h e r o t a t i o n of t h e centrifuge rotor) on [(r/ro) s - 1] is linear [10]

(C/2e),/o 7

/ 2 2/2//~/0z

/°l--

o

/

o

0

28~

•

•

•

0

I2 -~* 08 o

l

O~-

let.to5 I lO

FTG. 1. D e p e n d e n c e of ~/25 on r~t a n d r'5 of P P P f r a e t m n s m D M A A (•); D M F A - t - 0 25 L1C1 (2); DM_FA (3, 4). C o n c e n t r a t m n , g/dh 1, 3--0.03; 2, 4--0-02; a - - c o r r e c t i o n for dlstfllat m n is 1/3 of distillation tune; b - - c o r r e c t m n is d e t e r m i n e d f r o m t h e d e p e n d e n c e of In r on 5. I f for P P P fractions t h e correction for dlstfllatzon is d e t e r m i n e d b y t h e first two m e t h o d s , t h e dependence of ~m/25 on r~5 will be of t h e t y p e s h o w n m Fig. l a a n d t h a t of ln(r~/ro)/Cos5 on [(r~/ro) ~ - 1] will be non-linear. I f t h e correction is i n t r o d u c e d ~ecorchng to d e p e n d e n c e of In r (and n o t ]n rm) on 5 or b y t h e t h i r d m e t h o d , d e p e n d e n c e of ~z]25 on r~5 of P P P f r a c t i o n s will be linear (Fig lb). D e p e n d e n c e of In (r/ro)/Co25 on [(r/re) s - 1] will also be hnear. Tn e v e r y case, it is definitely m o r e correct to d e t e r m i n e t h e correction for dlstfllatzon f r o m t h e d e p e n d e n c e of In r on 5; however, for P P P f r a c t m n s tins is absolutely essentzal o w i n g to t h e considerable polydmpersmn of fractions, Mw/M~= 1 4 on an average [11] F~gure 1 shows t h a t t h e a c c u r a t e calculation of t h e correction for d~stfllatlon results m t h e d i s a p p e a r a n c e of t h e g r a d i e n t f r o m t h e d e p e n d e n c e of ~2/2t on r~t w i t h low v a l u e s o f r~5. A provzslonal e o r r e e t m n (difference b e t w e e n various m e t h o d s being ~ 100 see) has s u c h a considerable effect on t h e d e p e n d e n c e of ~2/25 on r~5 (r~5) n o t only because of t h e relatzve d u r a t i o n of chstdlatmn, b u t also because of t h e strong (quadratic) dependence of ~ on 5.

2638

I. K. N~ra~Asov

Resulting straight lines in Fig, lb, correspond to the Bresler and Frenkel equation [1, 12] derived independently [13]

~o,p2~lt

=D÷

2

(3)

E q u a t i o n (3) was derived assuming that the extension of sedunentatlon b o u n d a r y was independent owing to daffumon (D is the weight average diffusion coefficient) and polydispersion (p2 is the drsperslou of dmtribution q (s)) and enabled us to d e t e r m i n e / ) from intercepting the dependence of ~/2t on r2t and the value o f p from the gradient. I n most cases the sedimentation of P P P fractions was not oomphcated b y the concentration dependence of sedimentation coefficients a n d / ) and p values of fractions could be determined from formula (3). Equation (3) was derived by the summation of the dlspers,ons of distributions as a result o f polydlspermon and chffuslon. From theoretmal conclusions [4, 5] the standard deviation of the sedimentation boundary m a y be dweetly calculated. FU]lta [5] obtained an asymptotm solution for the equation of b o u n d a r y extensmn during sedimentation of a polydlspersed material for the case

s.D=eoust----K

(4)

(s----St.~),

which takes the form

q*(S,t)=q(s)--

--~

s .J

16,'ds k

s~

s'

s'

~

(5)

whereupon

(sa)

--_co4ro~t/4K Changing to a new variable x = r - - r o , from equation r=roe °'8t we obtain

x ~_eo2rotS,

(6)

a n d the apparent drstrlbutloa according to S at t moment of tune q* (S, t) is related to the distribution of q (x) with the ratio

q* (S, t) = ~ r 0 ~ (x)

(7)

Conmdermg equations (1), (5)-(7), &seardmg the term 0(1/r a) m equation (5) and assuming t h a t the sedmaentmg polymer does not contain a noticeable proportion of low molecular weight fractions, 1.e. conditions q(0)= 0, q'(O)= 0 hold good, we obtain

~ = 2 t b + co~ro~p't'

(8)

The Gostmg equation [3] derived with the assumption that D = c o n s t for all polymer fractions, takes the form:

q.,,..<,.

F

1

4DaB

D / z 1 \F

4Da5

l

(,)

2

a, = 1 + (soU) + ~ (s~U)*+ 5

1

2

a s = 1-{- ~ ( s ~ t ) - I --

.

(9a)

Boundary extenslon during sedlmentatlon of polymer solutions

2639

G,3vedbeqg u~ 4

¢~Su).lo;

i

J

O g ~-

2 -

-c'--Cd

~

08

d4

set.10 -s i 0

I

I~

(#f)xld~ 8e0-i

2 0

FIG. 2

Fie 3

FIG. 2 Dependence of ~*/2~ on r~t of polyacrylomtrlle fractions m DMFA. M × 1 0 ' ~ 6 . 5 (1,~), 11, 2 (3. 4) and 20, 9 (5, 6); concentration, g/dl. 1 - - 0 099; 2--0 309; 8, 5--0 2, 4, 6 - - 0 1. FI~ 3 Dependence of sedimentation coefl~clents S at constant Q* on 1[~ of P P P specimen B m D M F A + 0 25~o L1Cl, c=0.05 g/dl. The numbers at the straight hnes are the Q* values. yx/O~

/0-

A Ix

-/0 < 20-

B

}

-/0~ FIG. 4 Dependence of Y(s) on Q for P P P specimens A and B. The second terms m square brackets of equation (9) represent not more t h a n a few per cent of the first, therefore they can be ignored Taking a l : 1 +s~o~t a n d proceeding as m the first case, we obtain 2Dr ~2= _ _ + ¢oiro2p~t~ 1 + s~o*t

(10)

When sco~t <41 formula (10) a n d formula (8) are ldentmal a n d both ratios are practically eqmvalent to equation (3). Some d]fference (r 2 and r~) m possibly due to the use of the approximate ratio (6).

2640

I.K.

N~.~R~SGV

Figure 2 shows the dependence of ~'/2t on r ' t of PAN fractions m I)MFA. Sechmentatmn of PAN m DMFA ~s characterized b y strong concentration dependence of sednnentatmn coeffle~ents ~-----s(c) [14]. I)urmg considerable hydrodynamic mteractmn the effect of autocompresmon predominates and the boundary narrows as time goes on (curves 5 and 6). When reducing hydrodynamm interaction retraction becomes lower (ca~ve d) Finally, w~th shght hydrodynamic interaction the boundary expands as a result of polyd~spers~ou and dnTus~on (straight lines 1-3), so t h a t dependences of ~/2t on r~t have a positive gradient.

Talcing into account boundary diffusion extension in raput sedimentation of P P P in d~methylformamide--0 25 g/dl LiC1. A method ~s proposed for considering daffus~on boundary expansmn m rap~d sedkmentatlon of P P P solutions m D M F A - - 0.25 g/dl LICI to obtain the actual dlstr~butlon q(s) of sedimentation coefflc~ents s using a single photograph w~thout the cumbersome procedure of extrapolation of apparent d~str~butlons q*(S, t) or (Q* (S, t) is the integral d~str~but~on) to infinite t~me.

0 I0 ~'

0/

0~

oX

~<

Q2~ 0

.-~ •

o,¢

×

0X

"% °o

O~

_

>~

0~

/0 08

eo

02

-

08

%x o °a<

O~

-

x~

_

>,D

I

2

02

,..?

#

a"

0/

$

/ 2

d~

,.9, ~°vedbe.,,Szun

,S, ,~vedbel,q un

FIG. 5. Integral &strlbutmns of Q(8) for PPP, specnnens A (a) and B(b): / - - o b t a i n e d by extrapolatmn, calculated from apparent dmtrlbutmns of Q*(S, t); 2 - 3 0 6 0 , 3 - 3 9 0 0 , 4--4200 and 5--6600 sec.

.As the refractive index increment of the polymer m thin solvent is 0.303 ml/g, it was possible to carry out sedimentation experiments with unfractlonated P P P samples when c = 0.05 g/all, the angle of the phase.contrast plate being 45 °. The conventional method [1, 2] of e h m m a t m g diffusion boundary extension, winch is fairly well known m practice, revolves extrapolation of apparent d~stmbutlons Q*(S, t) to infimte tkme 1/t-~0. Using the graph for sedLmentatlon eoelTicmnt~ • at constant Q* accordmg to 1/t we find the values s----St.~ b y interception and finally obtain the actual distribution Q(s) of se&mentatlon coefficmnts. As an example Fig. 3 illustrates the dependence of on 1/t for a P P P specnnen B m I ) M F A - - 0 25 g/d] L1C1, 0=0-05 g/dl. The rectllmcarlty of curves makes it reasonable to use F u j l t a ' s theoretmal conelusmns [5] for this system. By the integration of equation (5) m respect of s we obtain

Q*(S, t ) - - Q ( s ) -

LFq(')

w h e r e T is d e t e r m i n e d b y t h e r a ~ o (5a)

(') /' (')l

(11)

Boundary extension during sedimentation of polymer solutions

2641

The dependence of sechmentatlon coefficients on time, Q* being constant, is determined by the equation

S-~-ao--i-al/~ ~

(12)

where

[

1 1

ao=s,

q'(s)

a l : ~ s ~ sq(s)

]

(12a)

F o r the Gausslan distribution of q (s) formula (12) becomes

1/1

~.--sm\

where sm is the sed~mentahon coefficmr~t m the rnaxlmum of the curve, p is the dmperslon of Gausslaa dmtrlbutlon. I t is evident t h a t for a good solvent ratm (4) should be regarded as approximate, ~ e K _~s. i)

(4a)

F u j l t a [5] proposed to find the value of K by the following method According to equations (12) and (12a) the gradmnt of the curve o f S vs 1/t Y (s) is

L

(14)

Bearing m m i n d t h a t dQ~q(s)ds, we o b t a i n

K [q(so)l,

I Y ( s ) d Q = Z = ° J ' r ° ~ "L so

Qo

j

(15)

where Z m the area under the curve showing the dependence of Y ca Q, values of Qo, q(so), So correspond to Y(So)~O. F r o m experimentally deterrmned values of Z, q(so) and s o using equation (15) the value of K m a y be calculated Figure 4 shows dependence of Y on Q for two P P P speelmens The area under the curves ranging from 0 to I Is 2Z F r o m results a K value of (0 9 5 ~ 0 . 0 9 ) × l0 -Ig was derived for three unfractlonated P P P speelmens and this value was subsequently used m the cMculatlons We substitute the distribution Q*(S, t) or q*(S, t) chaxaetemzed by average eoefficmnt and dispersion p*

(p*),= ~ (s-~)~q*(S,0dS,

(16)

0 by the equivalent Gaussian dlstmbutmn with parameters s ~ = ~ ' and p - - p * ratio (13) m somewhat modified

S=s+~+~)

I n tins case,

(13a)

Using ETsVM " P r o m m " and equation (13a) sedunentatlon coefficients s were calculated from S values of apparent dastrlbutmns Q*(S) and the actual dmtmbutlon Q (s) determined. Figure 5 shows a comparison of integral dlstmbutmns according to sedlmentatmn eoefflelents demved by extrapolahor~ and the dlstmbutlon calculated by a single-point m e t h o d from apparent distributions Q*(S, t) by formula (13a) Some characteristics of comparable curves are tabulated.

2642

I.K.

lqEr~aASOV

Figure 5 and tabulated data mdmate t h a t as time goes on, agreement between "singlep o i n t " distributions and the extrapolation curve improves. Approximately 1 hr after the begmmng of the expernnent the discrepancy does not exceed 2-3 ~ winch is, apparently, quite adequate for practical purposes. CHARACTERISTICSOF DISTRIBUTIONSQ*(S) AND Q(s) oF P P P s P E c I ~ . ~ s A A~D B Speclinen

t, sec

p*, Svedberg units

p, Svedberg units

A

3060 3900

0 889 0 812

0 754

ErrOr:o~ Spec~o linen 52 18

B

t, see

p*,Svedberg units

p, Svedberg umts

Error,

4200 6600

0 352 0 315

0-238

2.4 15

I n view of their approximate nature, the inaccuracy of nntlal assumptions for the smglopount method appears to be self-evident; to achieve success with this method when calculating the dLffumon boundary extension of P P P solutions m D M F - - 0 25 g/dl L1C1, further information is requtred. As pointed out previously, the FUjlta theory [15] is concerned with sechmentation of moderately polydmpersed polymers m ~ solvents without a concentration dependence of sechmentatlOn coefficients, the value of K being const. The P P P specimens studied were moderately dispersed; distribution is within the range (0 < s < 6) Svedberg umts; their weightaverage molecular weight is (80-180)× 103; for all specimens p <1 Svedberg unit and the sedkrnentatlon experiments were carried out with rather dilute solutions when c = 0.05 g/dl, so t h a t the constancy of K values for these specimens cannot, evidently be regarded as fortmtous. As shown by the Table, the value of p* for specnnen B differs considerably from 1o, hov~ever, "single-point" distributions show satisfactory agreement vnth extrapolated chstrlbutlon. I t is mgmficant t h a t the second term m the right hand side of equations (13) and (13a) is a correctmn and the error m the p value (substituted by p*) is reflected to a lesser e x t e n t by the s values. I t is evident t h a t the more prolonged ~he centrifuging, the less is the difference between S and s, tv and p* and the more reason there is for using the single-point approxunatlon to calculate diffusion boundary extension. The method examined is apparently, most appropriately used for polymers of moderate molecular weight and polydlsperslon, when dn~fusmn boundary extension cannot be ignored. The use of this m e t h o d is, no doubt, of special interest for 0 solvents: m this case formula (4) is valid and conditions, wlnch are due to the use o f r a t m (4a) for polymers m good solvents, are absent

CONCLUSIONS (1) A study was made of the dependence of ~s/2t on r2t of poly-m-phenylene-lsophthalamlde fractions m chmethylformamlde (DMFA), D M F A q- 0.25 ~ L1C1 and dmaethylacetamlde. I f for comparatively short experiments lasting 40-45 m m the correction for distillation ~s introduced by adding 1/3 of the distillation tnne to the mau~ time, dependence of ~2/2~ on T~t are concave u~ relation to the abscissa axis. I f the correction is taken from the dependence of In r on t or by adapting the F u j l t a equation, a rectil~uear relation m derived for ~[2t on r~t which conforms to the Bresler-Frenkel equation. (2) F o r polymers which are characterized by an arbitrary dlstmbutlon function accordmg to sedimentation coefficients q(s) b u t have no noticeable number of low molecular weight fractions, using the l~ujlta and Gostmg equation ratios were derived for ~2 whmh coincide m practice with the Bresler-Frenkel formula.

Boundary extensmn during sedimentation of polymer solutmns

2643

(3) Functions of ~'/2t dependent on r~$ for polyaerylomtrlle fractmns m DMFA are characterized either b y a negative or positive gradmnt, according to the degree of hydrodynamic mteraetmn. (4) By replacement of apparent dlstmbutmn according to s e d u n e n t a t m n coefficmnts Q*(S, t) b y the eqmvalent Gausslan dlstrlbutmn using the F u p t a theory it is proposed to calculate dlffusmn b o u n d a r y extensmn from a single photograph (5) The use of this method of calculating dlffnsmn botmdary extensmn during sedlmentartan of unfraetmnated poly-m-phenylene-lsopthalamldespec,mens m chmethylformamlde -0 25 g/dl L1C1, c = 0 05 g/dl showed agreement between single-point dlstmbutmns and dlstnbuttons obtained by conventional extrapolatmn wlttnn the range of error of 2-3 ~ approximately after 1 hour's sedlmentatmn.

Translated by E SEMERE REFERENCES ]. V. N. TSVETKOV, V. Ye. ESKIN a n d S Ya. FRENKEL', Struktura makromolekul v rastvorakh (Maeromolecular Structure m Solutions). Izd. "l~auka", 1964 2 S.R. RAFIKOV, S. A. PAVLOVA a n d I. I. TVERDOKHLEBOVA, Metody opredelemya molekulyarnykh vesov 1 pohdlspersnos~l vysokomolekulyarnykh soyedmenn, Izd AN SSSR, 1963 3 L. J. G0STING, J. Amer. Chem. Soe. 74: 1548, 1952 4 T. HOl~fl_& and H. FUJITA, J Appl. Polymer Sel 9. 1701, 1965 5 H. FUJITA, Blopolymers, 7: 59, 1969 6 I. Ya. PODDUBNYI and V. A. GRECHANOVSKII, Dokl. $1~ SSSR 175: 396, 1967, J. Polymer gel. C23: 393, 1968 7 V.A. GRECHANOVSKII and I. Ya. PODDUBNYI, Soveshehame po sedlmentatslonnomu 1 gel'-khromatograficheskomu metodam anahza MVR pohmerov (Conference on Sedimentation and Gel-Chromatographm Analysis of MWD of Polymers). NPO "Plastpolnner", 1970 8 J. E. BLAIR and J. W. WILLIAMS, J. Phys. Chem. 68: 161, 1964 9 H. FUJITA, J. Amer. Chem Soe 78: 3598, 1956 10 T. KOTAKA a n d N. DONKAI, J. Polymer Scl. 6, A-2. 1457, 1968 11 I. K. NEKRASOV, Vysokomol. soyed A13. 1707, 1971 (Translated m Polymer Scl. USSR 13: 8, 1920, 1971) 12. S. E. BRESLER and S. Ya. FRENKEL', Zh. tekhn, fizlkl 24 2169, 1954 13 J . W . WII.I.IAMS, R. L. BALDWIN, W. M. SAI_TNDERS and P. G. SQUIRE, J. Amer Chem. Soe. 74: 1542, 1952 14 J. BISSCHOPS, J. Polymer SeL 7: 81, 1955