The sedimentation, diffusion and viscosity of polybutyl isocyanate solutions

Sedimentatiou, diffusion and viscosity of P B I solutions

655

6. K. ISHII~&WA, K. MXIrASAKA, M. MAYEDA and M. YAMADA, J. Polymer Sci. 7, A-2: 1259, 1969 7. T. SETO and Y. TAJIMA, Japan. J. Appl. Phys. 8: 166, 1969 8. A. COWKING, J. G. RIDER, L L. HAY and A. KELLER, J. Mater. Sci. 3: 646, 1968 9. A. COWKING and J. G. RIDER, J. Mater. Sci. 4: 1051, 1969 10. A. I~EIJ.ER and D. P. POPE, J. Mater. Sci. 6: 453, 1971 11. M. A. GEZALOV, V. S. KUKSENKO and A. I. SLUTSKER, Mekhanlka polimerov, 51, 1972 12. B. M. GINZBURG, 1~. 8ULTANOV and S. Ya. FRENKEL, Vysokomol. soyed. A13: 2692, 1971 (Not translated in Polymer Sci. U.S.S.R.) 13. V. I. GERASIMOV and D. Ya. TSVANKIN, Vysokomol. soyed. A l l : 2652, 1969 (Translated in Polymer Sei. U.S.S.R. 11: 12, 3013, 1969) 14. A. Sh. G O ~ and T. P. TANTSYURA, Vysokomol. soyed. AI0: 724, 1968 (Translated in Polymer Sci. U.S.S.R. 10: 4, 839, 1968) 15. Yu. V. BRESTKIN, B. M. GINZBURG, P. A. IL'CHENKO, K. B. KURBANOV, M. A. MARTYNOV, Sh. TUICH1EY and S. Ya. FREN~EL, Vysokomol. soyed. A15: 621, 1973 (Translated in Polymer Sei. U.S.S.R. 15: 3, 549, 1973) 16. V. 8. KUKSENKO, A. I. 8LUTSKER and A. A. YASTREBII~SKII, Fiz. tverd, tela 9: 2390, 1967; S. N. ZHURKOV, V. S. KUKSENKO and A. I. SLUTSKER, Fiz. tverd. tela 11: 296, 1969; V. S. KUKSENKO and A. 1. SLUTSKER, Fiz. tverd, tela 11: 405, 1969; V. S. KUKSENKO and A. I. SLUTSKER, Mekhanika polimerov, No. 1, 43, 1970 17. A. GU]NIF~ and G. FOURNET, Small-Angle Scattering of X-Rays, J. Wiley Inc., I~.Y.; Chapman and Hall Ltd., London, 1955 18. V. S. KUKSENKO, S. NIZAMIDINOV and A. I. SLUTSKER, Fiz. tverd, tela 9: 1966, 1967 19. A. A. YASTREBINSKII, Dissertation, 1966 20. R. HOSEMANN and 8. N. BAGCHI, Direct Analysis of Diffraction by Matter, North Holland Publ. Co., Amsterdam, 1962; R. HOSEMANN, J. Appl. Phys. 34: 25, 1963 21. Ye. L. GAL'PERII~, V. F. MXNDRUL and V. K. SM][RNOV, Vysokomol. soyed. AI2: 1949, 1970 (Translated in Polymer Sei. U.S.S.R. 12: 9, 2207, 1970) 22. R. BONART, Kolloid-Z. und Z. f'dr Polymere 194: 97, 1964

THE SEDIMENTATION, DIFFUSION AND VISCOSITY OF POLYBUTYL ISOCYANATE SOLUTIONS* V. N. TSVETKOV, I. N. SHTENNIKOVA, ~/[. G. VITOVSXArA,YE. I. RYUMTSEV, T. V. PEKKER, YU. P. GETMANCHUK, P. N. LAVRENKO a n d S. V. BUSHn~ High Polymer Institute, U.S.S.R. Academy of Sciences

(Received 26 June 1972) The progressive diffusion D, sedimentation coefficient So, intrinsic viscosity [t/] and the molecular weights M of a number of polybutyl isoeyanate (PBI) fractions were measured over a wide range of M. The experimental results were used to deter* Vysokomol. soyed. A16: No. 3, 566-574, 1974.

656

V . N . TSVETKOV e$ al. mine the structural parameters of the P B I molecular chain on the theoretical basis t h a t this could be regarded as rigid, ellipsoidal, stretched and coiled persistent chains. The projection of the monomer chain u n i t on the molecular axis gave 2=2=t=0"1 /~ and represents a flat c/s-configuration of the chain. The persistent length a of the P B I molecule was determined from the results of progressive diffusion using the progressive friction theory for coiled chains in the range of larger M ; it was found to be 500 A. The P B I molecular conformations were found to be similar to a slightly bent rod only in the range of small M (less t h a n 4 × 104), while the best model for describing the hydrodynamic properties at larger M was the coiled chain.

A ~UMB~R Of a u t h o r s r e c e n t l y g a v e m u c h a t t e n t i o n t o t h e s t u d y o f s t r u c t u r a l a n d p h y s i c a l p r o p e r t i e s o f p o l y i s o c y a n a t e m o l e c u l e s [1-10]. R e c e n t p u b l i c a t i o n s [5, 8] h a d s h o w n t h a t c o n s i d e r a b l e r i g i d i t y a n d o r i e n t a t i o n o f i n t r a m o l e c u l a r s t r u c t u r e is t y p i c a l for t h e p o l y b u t y l i s o c y a n a t e ( P B I ) m o l e c u l e s , a n d t h a t t h e c o n f o r m a t i o n a l p r o p e r t i e s o f t h e s e m o l e c u l e s c a n b e d e s c r i b e d b y a coiled c h a i n model. T h e s t u d y d e s c r i b e d h e r e d e a l s w i t h a series o f P B I f r a c t i o n s b y s e d i m e n t a t i o n , d i f f u s i o n a n d v i s c o m e t r y o v e r a w i d e r a n g e o f m o l e c u l a r w e i g h t s (M). The a i m was the e s t a b l i s h m e n t of correlations b e t w e e n the h y d r o d y n a m i c prope r t i e s o f t h e m o l e c u l e s a n d M , a n d also t h e u s e o f t h e s e c o r r e l a t i o n s i n t h e det e r m i n a t i o n of the m a c r o m o l e c u l a r c o n f o r m a t i o n characteristics. EXPERIMENTAL

P B I samples were synthesized as described before [1]. Fractions were produced b y fractional precipitation with methanol from the tetrachloromethane solutions. The range of M was broadened b y reducing the mol.wt, of one sample by ultrasonic irradiation before fractionation. The viscometric determinations were carried out at 21°C on tetrachloromethane solutions in a capillary viseometer. [7] of the fractions with M ~2.8 × 10~ was determined from the shear stress b y extrapolation of [7] as a function of velocity gradient g to g--~0. Tetrachloromethane was also used as solvent in the diffusion and sedimentation tests. The average increment of the refractive index of system PBI-tetrachloromethane, An/ztc, was 0.045 ±0.005 cm3/g. The progressive diffusion coefficient D of the P B I fractions was determined on the polarization diffusimeter [11, 12]. Where relatively small M fractions wore used (M < 9 × 104) a well defined initial b o u n d a r y surface was obtained by using an undorlayering diffusion cell with a modified chamber. A special aperture was drilled into the side wall of this chamber through which the diffuse solution-solvent b o u n d a r y layer was removed. The interferometric diffusion curves were processed b y measuring the surface areas of the greatest ordinates. The linearity of the plots shown in Fig. 1 permits t h e use of the slopes to determine D for the P B I fractions; these values are contained in Table. The study of the diffusion coefficient as a function of concentration on one of the fractions with M = 7-67 × 105 showed D to be almost independent of c in the range c × l0 s = 0-09-0.02 g/cm 3 (Fig. 2), so that the value of D for each P B I fraction could be determined from a single concentration lying within this concentration range. The sedimentation coefficients S wore established in a G-120 (Hungary) ultracentrifuge fitted with a polarization interferometer attachment. The density difference between the polymer, pp, and the solvent, p~ (tetrachloro-

S e d i m e n t a t i o n , diffusion a n d v i s c o s i t y o f P B I s o l u t i o n s

657

THE HYDRODY~AI~IIC CHARACTERISTICS OF P B I FRACTIONS Ilq TETRAOHLOROMETHANE Fraction,No.

[e] × 10-2

D × 107

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

85 63 36 29.3 27.4 26 24 15 12-7 10 5.0 3.8 1.40 1.12 0.85 0.48 0-10

0-22 0-32 0.57 0.53 0.73 0.68 0.75 0.68 1.25 1.30 1.47 2-3 5-0 5.65 7.2 7.4 17.1

- - S o × 10 la M s D x 10 -5 5.4 4-35 3.7 3.7 3.45 3.4 3.8 3.05 2-80 2.56 2-34 2-11 2.4 2.0 2.1 1.5 1.3

13'8 7'67 3"66 3-93 2"66 2-82 2-86 2'53 1-26 1.11 0"897 0"52 0"271 0-20 0'165 0"114 0"038

Ao × 10 x°

p

H × 10 -a

3.55 3.84 4.42 3'93 4.65 4.34 4.68 3-49 4.82 4.43 3.7 4-43 5-5 5.25 5.72 4.3 4.15

500 400 300 260 200 290 235 178 160 142 97 83 47 40 34 24 10

9.95 7-05 4.55 4.23 3.12 4.07 3-56 2.84 2-092 1.856 1-339 1.013 0.554 0.450 0.379 0.261 0.107

m e t h a n e ) , w a s small; pp •PS* T h e s e d i m e n t a t i o n r a t e s w e r e t h e r e f o r e d e t e r m i n e d Lri a cell p a r t i t i o n e d b y a n artificial b o u n d a r y surface a n d t h e r o t a t i o n v e l o c i t y o f t h e r o t o r was 40,000 r p m . F i g u r e 3 s h o w s t h e d e p e n d e n c e o f A log x0 o n t i m e for P B I f r a c t i o n s o l u t i o n s ( x 0 - - r a d i a l c o o r d i n a t e o f t h e f l o t a t i o n b o u n d a r y p e a k a t t i m e t). T h e slope o f t h e lines d e t e r m i n e s t h e r e s p e c t i v e s e d i m e n t a t i o n coefficient.

1o,'/z

I /

-

"

#/

~

12

7

o

~i

0

8

18

I

10 l

I

l

50

I00

150

~

I

I

l

tO0

l#O

290

,.

t , / 0 - a sec

FIG. 1. The dependence o f 1 / K ~ 4 D t on t i m e t for P B I fraction sloutions in tetraeldoromethane. The numerals against the lines are those of the fractions here and in Figs. 3a and 4; 0 - - ~ f r a c t i o n a t e d sample.

V. ~ . T s v ~ o v

658

et al.

T h e fractions with ~ large _~f were used to study the dependence of the sedimentation coefficients a t finite fraction concentration S* on c (Fig. 4). The slope of the curves became less steep in the concentration range c > 0 " 0 5 × 10 -2 g/dl, probably due to interinolectflar

,zo 7, cm sec O.q[

o

0 "2

c,

i

i

Z

I

g

c,lO z, g/d/

Fzo. 2. The dependence of D on the solution concentration of t h e P B I fraction with 2 ~ = 7-67 X I05.

0"01 g

1

!

,

0

0

i

I

~\

I\ Z

Sz? Tz'me~rain

I

I

20

zlO

T/me, rain

L

,

I

log

6O

FIG, 3. a" J log xo as a £mmtion o f f l o t a t i o n tLme for P B I f r a c t i o n solutions m tetrachloromethane; b: as a for fraction 1 at c, g/d]: 1--0.011; 2--0.015; 3--0-022; 4--0.032; 6--0.048.

Sedimentation, diffusion and viscosity of P B I solutions

659

reactions, as [7]" c ~ 1 for the large M fractions at these concentrations. The sedimentation constants S0 were calculated for all the fractions taking into consideration the concentration dependence, which can be d e s c r i b e d - - 1 / S * ~ 1 / - - S o ( 1 ~ 7 [7] c)[14], for which the average experimental y=0.3, using the linear p a r t of the curves; these values are given in the Table. The determination of the sedimentation coefficient for the fraction with the lowest M (S0--=-- 1.3 × 10 -zs) was not reliable enough; the Archibald [15] m e t h o d was therefore used to determine M for this fraction. The result, i.e. Mw~-3"8× 108 is close enough to t h a t calculated for this fraction, i.e. M s D ~ 4 - 3 × 103.

0: 0"4 [

[

:.6 0.4 O..5 0.3 0"4

0"~ O.O2

0,04 c,,~/d:

FIG. 4. The dependence of l/S* on the P B I fraction concentration in the tetrach]oromethane solution. Only v e r y small amounts of the P B I fractions were available for the determinations, so t h a t t h e concentration dependence was examined after consecutive dilutions made directly in the sedimentation cell. The exact concentration was determined from the contour height of the interference line which complied with t h e theoretical shape [13].

F i e . 5. Flotation interferogram of t h e P B I fraction 2 solution in tetrachloromethane; c = 0 . 0 5 × 10 :~ g/dl. Concentrations up to 0.01 × 10 -* g/era 3 could be determined by this method with an up to 1 0 ~ accuracy (equivalent to a rectangular interference c o , t o u r line having a 0.1 distance between the wedge lines). Th~ values D and So were used to determine the mol.wt, of the fractions by means of the Svedberg formula at the specific partial volume ~=0.898 of the polymer. The values of M, and also of t h e parameter Ao=--Dvlo(M[~])tT -z, are given in the Table (~0--solvent

V. N. TSV~TKOV e~ a/.

660

viscosity). A0 agreed within experimental error for the various fractions, bu~ the averages, i.e. A0=4.5 x 10-I° were slightly higher than those normally obtained for flexible coiled molecules, i.e. A0~3"4X 10-~° [11]; this had also been found earlier during the study of the hydrodynamic properties of a synthetic polypeptide-poly-y-benzyl glutamate [16]. RESULTS

The experimental values of[t/], D a n d S o were plotted against M using logarithmic coordinates (Fig. 6). For M in the range 4 x 103 to 1 x 10 ° the dependences of [t/I, So a n d D can be described b y the equation [t/]-=K,M%=4.0 x 10-' M T M

(1)

D=K~M-b-~8"7 x 10 -4 M -°'Ts

(2)

--So=KsMl-b= 1.5 X 10 -14 M °'~5

(3)

The large exl0onents in the equations for [t/] a n d D (~= 1.26, b-----0.75) c a n n o t be explained as due to steric effects [17] a n d indicate large rigidity (as well as easy permeation) of t h e P B I molecular chains a n d also conformational differences

/og//i) d-

-y2 _

--7

3

-/3 0 3.5

¢.5

,~5

io/M

Fro. 6. The dependences on log M of: /--log [t/]; 2--log D; 3--log So for PBI fraction solutions in tetrachloromothano. f r o m a Gaussian coil shape. Curvature of the log [t/]-----f (log M) plot to a less steep slope can be seen in the larger M range, i.e. a decrease of exponents a n d b. The explanation could be a decrease in permeability of the macromolecules, i.e. b y an increase of the intramolecular h y d r o d y n a m i c reactions a n d lengthening of the molecular chain. The s t u d y of the h y d r o d y n a m i c properties of P B I molecules over a wide range of M (400-fold) enabled us to demonstrate the existence of some molecular flexibility a n d also to get some q u a n t i t a t i v e information of molecular parameters such as the length projection of the m o n o m e r chain unit on the molecular chain a n d the diameter of the latter.

Sodimentation, diffusion and viscosity of P B I solutions B y m o d e l l i n g t h e P B I molecule as rigid, fiat, eUipsoidal bodies v o l u m e a n d mass, w h i c h are n o t s o l v a t e d b y t h e solvent, t h e values t h e solution can be used to d e t e r m i n e t h e d i m e n s i o n s of t h e e q u i v a l e n t T h e intrinsic v i s c o s i t y o f t h e a b o v e solutions can be p r e s e n t e d in t h e f o r m [18]:

_N" V v in which its length [19]. The *ermined

(.e) ,

661 of e q u a l of [~/] for ellipsoid. following

(4)

v is the v o l u m e of ellipsoid, v (P), a function of ratio P = H / d , i.e. of H to its diameter (width) d, which is tabulated as the Simha function value of v for the ellipsoid, in the absence of solvation, can be defrom the partial specific v o l u m e ~ of the substance in solution, i.e.

v = ~ (M/NA).

8

2

(5)

10

14 Z*/O "~

e/Z,

I

Z,,lO-a 0

0

o

N

o

2

I 8

I

d

/5"

o

I

/2

Z*IO -z

f

°°

I Z.~-a FIG. 7. The dependences on polymerization efficiency Z for P B I fraction solutions in tetrachloromethane off a--H/Z acc. to eqn. (6), b--H/Z ace. to eqn. (8), c--as b on a scale up to Z < 1 0 8, d--Hip.

662

V . N . Tsv~xov e$ al.

Using eqn. (4) the viseometric results thus allow determination of the axial ratio p of the equivalent ellipsoid if M is known. The respective values of p were found to be in the range 10-500 for the P B I fractions studied (see Table). The values ofiv make it possible to calculate the axis H of the equivalent ellipsoid from the following formula in which use is made of eqn. (5):

I-I3=6~v/~

(6)

The ratio H/Z becomes smaller as the polymerization efficiency Z increases, as Fig. 7a shows, and this indicates quite a considerable chain flexibility and deviation in shape (with increasing Z) from a rod-like one. The critical H/Z in the range of small M gave (H/Z)z~o-----t,which is the projection of the monomer unit length on the molecular chain; this was 2 A. The plot of Hip against Z (Fig. 70) and its extrapolation to Z - , 0 permits the estimation of d of the equivalent ellipsoid, i.e. d = 10.5 A. The progressive friction coefficient of stretched rotation ellipsoids (p>10) is correctly described b y

kT

3~.oH

f---- D = l n 2 p

(7)

By assuming t h a t H = 2 Z for such bodies (this assumption will be met with greater accuracy when Z becomes smaller), and by considering eqn. (7), we get

A- - -,.- u r n H ,. kTln2p ----rim -----z-*0 Z

(8)

z-~0 3=t/oZD

The value of Io present in the right hand side of eqn. (8) can be determined from the viscosity. Figure 7b shows the general progress of H/Z= p (Z) to be identical with that of the curve shown in Fig. 7a, which had been plotted from the viscosities. H/Z = q (Z) will give A= 2 A when Z-* 0 which is in agreement with t h a t found from the intrinsic viscosities. The fact t h a t 3 points fall outside the range Z = ( 2 - 3 ) × × 10 ~ seems to be due to a considerable error in determination of M (of coefficients D and So), especially in this range of Z. Values of 2 and d found by using the stretched, flat ellipsoid as a P B I molecular model ( p > 10) is equivalent to a flat cis-eonflguration of the P B I polymer chain [5, 8]. Quantitative evaluation of the equilibrium flexibility of the molecular chain can be made on the basis of the dependence of progressive friction (sedimentation and diffusion) on M for PBI. The hydrodynamics of the macromolecular chains must be taken into account here as it gives consideration to permeability. In the most adequate form these properties will be described by a coiled chain model [21]. The use of the latter when L > 2-2 A (L is the contour length of chain and A the length of statistical segment) will give the following equation [22]:

( ~--~----) (1.843/3gt/oNA) (MILA)*M~"+(3~tI°'N"O-'(MIL)[InAId--1"43](9)

\

Sedimentation, diffusion and viscosity of PBI solutions

663

Plotting experimental values of D M / R T against M ~ gives a straight line whose slope permits determination of A (Fig. 8a). The experimental points fall on a straight line for a wide range of M, but there are deviations at small M ( M < 0 . 4 x 105) (Fig. 8a, curve 2) because L>>2.2 A does not apply to this range. By using D found for larger M and the slope of line 1 in Fig. 8a was found as A (1000@50) /ix for M / L = 5 . 0 S X 109 0 , = 2 A). This value of A is equivalent to 500 monomer units per segment (8=500). The distance from zero at which line 1 intersects the ordinate makes it possible to estimate the diameter of the P B I molecular chains as 6 •. For the range of L < 2 . 2 A (for d<
M a

L

--0.002 ( ~ - ) [ ( - ~ - )

a

(

L

d ['L i

--4 ]@(-~-)

-1

"~ -.)+0

/d \~[/L\ 2

d

d

e

1 d

[[email protected](~-)@0.040(-~-) --0"006(~-)

]}(10)

3

Curve 2 in Fig. 8a was plotted by using A = 1000 and d = 6/ix, which were determined in the large M range. The experimental points were grouped around curve 2 showing t h a t the molecular conformations of P B I are similar to those of a slightly bent rod in the range of fairly small M. According to theory [23] the molecular chain rigidity (i.e. value A or s) can be estimated by other independent methods, i.e. from the intrinsic viscosities. For a coiled chain modelled on Gaussian chains and fairly large M , t h e theory gives the following function [23]:

.

at

is

A

,

"

2

1

1

(,1,

in which ~ - - F l o r y constant. As Fig. 8b shows it is almost impossible to get a useful slope from the curve plotted in the range of M used; for M~<2.5× 105 the experimental values of J//[t/] do not fulfil the condition of eqn. (11) [23]. The theoretical function for quite rigid molecular chains, such as those of P B I and small M, will apply when chain length L < A , but much larger t h a n diameter d (the P B I fraction with the smallest M has L ~ 10d and the conformation of a slightly bent thin rod) [23] 2uNAM 2 1 [- 3 1 In (L/d)-- 1.03 [t/] = 4 5 (M/L) a" In ( L / d ) - 1.03 L 4 @ 4 - " In (L/d)--1.7

M ] 16A~/L)

(12t

The above equation can be transformed to

( L \3

1 /' L \~

(12a)

i

864

V. N, TSVETKOV 6t a/.

in which

L In ~

[,1]

1.03

45

Y= M ~" 2~2YA " 3 4 3 X~

(L/d)--l.O3

1 In

4 in (L/d)--l.7

1

n-d---1.03

M

-4--~ 4

L l n - ~ - - 1.7

16

The initial ordinate of function y = f (x), which equals lira y = ( L / M ) 3 makes 0

it possible to determine parameter ~, and the initial slope which equals 1/A (L/M) 4, segment length A. Function y = f (x) which was plotted from experimental values of [~], M and L=,~Z, is illustrated i n Fig. 9.

:f

/2 2

H//[q]'10-~

o

2-gOo

o

4 / I

I

N

/2 M ½ . / O "z

4

g

/2 kl~.fO "z

FIG. 8. ~" DM/RT, b: M/[tl], as functions of Mt for PBI fraction solutions in tetrachloro. methane. The processing of the .points shown in Fig. 9 b y least squares method gives lira y = ( L ] M ) 8 = 10.6 × 10 -80 cm 8 and the slope (L]M) 4. 1]A = 1.1 × 10 -3~ cm 3. X--*0

Parameter ~ calculated from the initial ordinate, i.e. 2=(2.2=~0.1) A, agreed g, lOz

I0

30

JO

7o

FIG. 9. The dependence of y on x (see eqn. (12a)) for PBI fraction solutior~ in ~trachloromethane.

well with t h a t found earlier ~ = 2 A based on [~] and progressive friction when t h e precise formulae of Simha and Perrin are used. As to length A, it is shorter

Sedimentation, diffusion and viscosity of PBI solutions

665

b y a f a c t o r of a t least 4 w h e n o b t a i n e d f r o m t h e slope of t h e y = f (x) function, i.e. A=220=[=40 A, c o m p a r e d with t h a t f o u n d f r o m progressive friction. Such a large difference in t h e result f r o m progressive friction a n d v i s c o s i t y c a n n o t .be caused b y e x p e r i m e n t a l errors. This f a c t clearly illustrates the ina d e q u a c y o f t h e molecular c h a i n viscosity t h e o r y for t h e r a n g e o f small M w h e n t h e molecular c o n f o r m a t i o n is similar to t h a t of a slightly b e n t rod. The progressive friction t h e o r y is m u c h more promising u n d e r these conditions a n d is preferred in studies of rigid molecules such as those of p o l y a l k y l i s o c y a n a t e s . The h y d r o d y n a m i c s o f t h e P B I molecules for a wide r a n g e of M t h u s show t h a t the c o n f o r m a t i o n of t h e P B I molecules is close to t h a t of a slightly b e n t r o d o n l y in t h e range o£ small M ( M < 4 × 104) a n d t h a t t h e chains are flexible at larger M. The best model for describing t h e i r c o n f o r m a t i o n a l properties is t h e P o r o d coiled chain. A q u a n t i t a t i v e m e a s u r e of t h e rigidity of such a c h a i n is the persistent length a-:A/2~500 A. The a-value agrees well w i t h t h a t obt a i n e d in some of t h e earlier studies [5, 8, 10]. Translated by K. A. ALLEN REFERENCES

1. 2. 3. 4. 5.

6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23.

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