Determination of the polydispersity of polymers in different chromatographic regimes

1018

A.A. GORBUNOVand A. M. SKVORTSOV

REFERENCES 1. S. N. KOIKOV and A. N. TSIKIN, Elektricheskoye stareniye tverdykh dielektrikov (Electrical Ageing of Solid Dielectrics), Leningrad, 1968 2. M. A. BAGIROV, V. P. MALIN and S. A. ABASOV, Vozdeistviye elektricheskikh razryadov na polimernye dielektriki (Effect of Electrical Discharges on Polymer Dielectrics), Baku, 1975 3. Noveishiye metody issledovaniya polimerov (Latest Methods for Investigating Polymers) (Ed. B. Kee), Moscow, 1966. 4. A. CHARLESBY, Yadernye izlucheniya i polimery (Nuclear Emissions and Polymers), Moscow, 1962 5. V. V. KOCHERVINSKII, V. G. SOKOLOV, B. N. ZAIKOV and Yu. V. ZELENEV, Vysokomol. soyed. A19: 1843, 1977 (Translated in Polymer Sci. U.S.S.R. 19: 8, 2111, 1977) 6. N. I. AFANAS'EVA, G. B. VITSKUDEL' and G. I. ZHIZHIN, Zh. prikl, spektroskop. 21: 276, 1975

PolymerScienceU.S.S.R.Vol. 29, No. 5, pp. 1018-1025,1987 Printedin Poland

0032-3950187$10.00+.00 © 1988 Pergamon Press pie

DETERMINATION OF THE POLYDISPERSITY OF POLYMERS IN DIFFERENT CHROMATOGRAPHIC REGIMES* A. A. GORBUNOVand A. M. SKVORTSOV Leningrad Chemicopharmaceutical Institute

(Received 14 October 1985) A method is proposed for evaluating the MD not requiring preliminary calibration against standards and based exclusively on the use of parameters of the recorded chromatogram. For this the authors employ the theoretical dependence relating the slope of the calibration curve to the distribution coefficient. The influence of adsorption effects on the widening of the chromatograms is considered. A new way of evaluating the instrumental widening is proposed using the fo~m of the chromatograms close to critical conditions.

ONE of the important characteristics of polymers is their polydispersity. A full idea of polydispersity is given by the MD. The quantitative characteristic of polydispersity is the dispersion ¢r2 (or the standard width aM) of the MD function. Usually as parameter of polydispersity the ratio of the different moments of the MD is used, for example, the magnitude U=Mz/Mw [i]. For not too wide MDs a link exists between the parameters U and aM [2] U,,~ 1 +(aM/Mw) 2 * Vysokomo]. soyed. A29: No. 5, 920-925. 1987.

(1)

Determination of polydispersi~yof polymers

1019

An effective method for determining polydispersity is GPC [3]. The usual procedure for determining polydispersity by the GPC method is as follows [2, 3]. At first calibration is carried out using standard narrow-disperse samples (fractions) and a calibration curve is plotted, i.e. the link between the MM of the polymer investigated M and the retained volume VR = Vo + Vp'K(Vo is the free volume of the chromatographic column; Vp is the volume of the sorbent pores; and K the distribution coefficient). The experimental chromatogram is, where necessary, corrected by excluding the effects of "instrumental widening" unrelated to the MD of the sample. Then using the calibration the chromatogram is recalculated, i.e. the retained volume distribution by the MD function, the moments of this function are calculated and the polydispersity parameter determined. An inconvenience of the procedure described is the need for preliminary calibration and the attendant requirement for well characterized narrow-disperse standards. The present work proposes a way of evaluating the magnitude U not requiring preliminary calibration and based ecxlusively on the use of the parameters of the chromatogram. We would firstly note that the parameter U may also be evaluated without calculating the MD function. Let the test polymer be characterized by .~ = M~, and the standard width of the MD function ¢M and the corresponding chromatogram by the mean value Va and the width a v. We shall consider that within the limits of av the link between VR and the MM may be approximated by a straight line. Then

av~ aM~ Vp aal_~ M aM/if'I,

av~ ~

(2)

whence follows

Thus, to evaluate the polydispersity it suffices to know the width of the chromatogram rTv, the pore volume Vp and the magnitude OK ~=31nM

(4)

In chromatography the calibration dependence of Va on the logarithm of MM is usually employed and, therefore, the magnitude ¢ characterizes the steepness of the calibration curve. Below we show that the magnitude ~, may be linked with the distribution coefficient K and in this way from formula (3) determine U using only the parameters of the chromatogram itself. Case o f GPC. As is known [4] the value of the distribution coefficients of flexible Gaussian chains in the GPC region is essentially determined by the value of the parameter 9 =R/d, i.e. by the ratio of the mean radius of inertia of the macromolecule R = ~,~-~]

to the pore radius d (b is (he size of (he segment; Mo is its MM). The depend-

1020

A.A. GORBUNOVand A. M. SKVORTSOV

ence o f K on the f o r m o f the pores of the adsorbent (slit, cylinder, sphere) is manifest only in the value o f the numerical coefficients. Below f o r the sake o f determinacy we shall speak o f the pores o f slit-like form. In [4] exact relations were obtained for K(g) and also approximate f o r m u l a e 1-

'

for

"J"g

d>R

2

(5)

Since f o r Gaussian chains

OK - -

O OK ,

-

~=01nM

2 0#

[ (I-K)/2

for

(6)

then f r o m relations (4)-(6) follows

d>R

,' 8 \

(7)

Figure 1 presents the dependence o f ]~1 on K (the b r o k e n lines indicate calculation f r o m the f o r m u l a (7) and the continuous f r o m exact formulae for K(g) [4]), for c o m p a r -

IrJl 0"3

t

\

0"3

03 0"I

0.5

1.0

I.S K

0

O.q

0.8

K

FiG. 1 Fro. 2 Fie;. 1. The magnitude [~{ characterizing the width of the chromatogram as a function of the distribution coefficient K in GPC conditions for slit-like (1) and cylindireal (2) pores. Change in K due to change in the parameter g = Rid. Approximate relations calculated from formula (7) are shown by a broken line. FIO. 2. The magnitude [~l as a function of the distribution coefficient K in conditions where the increase in K is due to rise in the adsorption interactions of the units with the pore walls, g = R/d= 0"1 (1); 0.3 (2); 0.5 (3); 1 (4) and 10 (5). Circles denote the minimal values Km,. corresponding to GPC conditions, i.e. suppressed adsorption.

i)etermination of polydisperaity of polymers

1021

ison it gives the dependences for slit like and cylindrical pores. It will be seen that the general character of the function ~,(K) persists for pores of different shape (the maximum difference does not exceed 15 ~o) and for slit-like pores is well described by the formulae (7). From the analysis made it follows that to evaluate the polydispersity parameter it suffices to have a gel chromatogram, know the parameters of the chromatographic column Vo and Vp and use formula (7). According to Fig. 1 for a given sample the maximum widening of the chromatogram in GPC conditions may be achieved by using adsorbents with pore width 2d~rcR when the distribution coefficient becomes equal to 8 K=~e-l~0.3

Determination of polydispersity in conditions of weak adsorption of the units on the pore walls. Above we considered the case when change in the distribution coefficient is linked with change in M M of the polymer or variation in the pore width of the adsorbent. Let us now consider the situation when the adsorbent and polymer sample are fixed and the polymer-adsorbent interaction changes and minor forces of attraction (adsorption) begin to act between the chain units and the pore surface. Such conditions are experimentally realized either on worsening the thermodynamic quality of the solvent or in mixed solvents [5-8]. By changing the composition of the solvent or temperature it is possible within wide limits to vary the distribution coefficient for fixed MM of the sample. A theoretical description of the influence of adsorption effects on chromatography of polymers is contained in [9]. It is shown that in the general case the distribution coefficient depends on two parameters ,q = R/d and ). = - d/H and is equal to 22 2 K = .~= 2 (22 ~-)~+ ~2) exp [-- (g~n) ], an

(8)

where ~, are the roots of the characteristic equation; ~.tan ct=2; H is the correlation length characterizing the adsorption unit-pore surface interactions [9]. In the case of" GPC H = 0 and formulae (8) pass into the known relations (5). As the adsorption interactions between the chain units and pore walls intensity the distribution coefficient rises but at first remains less than unity so that the order by MM of outflow of the components remains similar to the order of outflow in GPC. We shall call this region of conditions precritical. Its boundary for narrow pores is the condition - H < d ( f o r K < 1 the correlation length H is negative) and for wide pores the condition -H d and for wide pores by the condition - H > R. At the most critical point Kcr=l and H~ 1 =0. In these conditions all the chains

1022

A.A. GORBUNOVand A. M. SKVORTSOV

regardless of their MM emerge with the same retained volume, which has been repeatedly observed experimentally~[5, 7, 8]. At K > 1 the MM order of outflow of the components changes to the opposite and the magnitude H becomes positive. If, however, H>d and accordingly K ~ 1 the general patterns characteristic of the near-critical region persist. With further increase in adsorption the magnitude H diminishes and the distribution coefficient becomes appreciably more than unity. Because of the slow establishment of equilibrium these conditions are inconvenient for chromatography of polymers. Let us at first consider the general realtions. Using formulae (6) and (9) we have "2 ,2 ~ exp I-(#~t,) 2]

(9)

which allows us, knowing ~1and g, to determine from formulae (3), (4) and (9) the polydispersity parameter U. As in the case of GPC it is convenient to examine the dependence of ~, on K. We shall take the magnitude # as constant and begin to increase the distribution coefficient by increasing H (this is achieved experimentally by varying the external conditions: changing the composition of the mixed eluent; temperature, pH, etc.). The function ~(K) plotted from formulae (8) and (9) is depicted in Fig. 2 for several # values. The empty circles denote the smallest values of the distribution coefficients Kmi, which may be realized for the given magnitude # =Rid. These Kmi, values correspond to conditions of total suppression of adsorption effects, i.e. the GPC condition. The curve on which these points lie evidently matches the curve in Fig. 1. If the # and K values are known then from Fig. 2 one may evaluate how far are the experimental from GPC conditions. As may be seen in the general case ~uis a function not only of Kbut also of g, i.e. depends on the ratio of the chain and pore sizes. According to Fig. 2 for # > 1, i.e. for narrow pore adsorbents (or sufficiently long macromolecules) ~' is determined, in fact, by only the magnitude K (Fig. 2, crave 5). In this case to obtain the polydispersity parameter U it suffices to known only av and K as was done in the preceding section for GPC. Since change in K in Fig. 2 was achieved by varying the adsorption capacity of the chain units with constancy of the mean dimensions of the macromolecule and no change in MD, the non-monotonicity in the behaviour of ~t(K) denotes non-monotonicity of change in the width of the chromatograms with change in eluent or temperature. The reason for such non-monotonicity is that on naxrow pore adsorbents d < R in GPC conditions the distribution coefficients are small (left part in Fig. 2). In fact, the macromolecules emerge almost at the limit of exclusion so that the width of the chromatograms is small. On the other hand, in the precritical region, according to the results of [9], the influence of adsorption of the units on the walls of a narrow pore is manifest as the effective widening of the pore by the value IHI ,.8/'

d

\

I-

rc

R

2

Determination of polydispersity of polymers

1023

Accordingly, the macromolecules move from the limit of exclusion and the chromatogram widens as K increases. In this region ,., 8 ]y/l~Kln(~),

(11)

so that [~u]rises with increase in K. With further increase in the forces of adsorption and the associated transition to the near-critical region the distribution coefficient becomes equal to K ~ e x p ( - 2 0 2 ) = e x p ( - ff--~)

(12)

Iwl gllnKI

(13)

and, consequently,

For K~< 1 the magnitude I~/] and, consequently, also the width of the chromamgram diminish with rise in K. At the most critical point (for Kcr =1) the width of the chromatographic peaks, as stated, ceases to depend on the MD of the sample and is determined only by instrumental widening. For K g I and H>d, i.e. in the near-critical region on the other side of the critical point, formulae (12) and (13) remain true and thus the width of the chromatograms again starts to grow. With further increase in K and passage to the adsorption region the width of the chromatograms lises and does so more sharply the smaller the value g = Rid (right part of Fig. 2). In the region K< 1 the maximally wide chromatograms with ~'m,x = e- 1 will, according to formulae (1 I) and (I 3), be in the precritical region for K=e-l~0'37

(14)

From the condition (14) follows the possibility of evaluating the polydispersity of the polymers convenient for serial analyses. In varying the composition of the solvent or temperature it is necessry to find conditions in which the width becomes a*. In these conditions /0"* X2 U~, l + ( ~ e ) (15) The conditions found and formula (15) remain valid for all molecules the size c~f which exceeds the pore diameter. Now, let us consider the situation for wide pore adsorbents. We shall consider the adsorbent wide-pore if the pore radius exceeds the characteristic size of the macromolecule (d> R). In the GPC region the distribution coefficient is given by focmula (3) and

1024

A.A. GORBUNOV and A. M. SKVOR'rSOV

~u by formula (7). According to theory [9] on passing to the precriticalregion the distribution coefficientso changes as ifthe size of the macromolecule fellby the value ]H l (16) Accordingly, 1 R

(17)

and does not depend on H. Consequently, in wide pores the adsorption effects at first do not change the width of the chromatograms. With further increase in K and transition to the near-critical region K ~ 1 the relations [9] R2 K ~ I +292=1 + dH (18) ~'~I-K

(19)

are fulfilled. This means that the width of the chromatograms begins to fall linearly with rise in Kand tends to zero at the critical point. On passing beyond the critical point (for K g 1) the formulae (18) and (19) remain true and consequently the width of the chromatograms rises linearly with K. The linear character of the widening of the chromatograms in the near-crit ical region allows us to determine the instrumental widening from the chromatographic data alone by extrapolating the width of the peak to critical conditions. Let us discuss some features and limitations of the method proposed. The idea is to replace the experimental calibration curve Va(ln M) by the theoretical function ~(K) linking the slope of the calibration curve with the distribution coefficient (Figs. 1 and 2). The Gaussian chain serves as a model of the macromolecule and it is assumed that the adsorbent used is uniform both in pore width and pore shape; the latter, by the way, is less important (Fig. 1). The applicability of the model of the Gaussian chain for describing the distribution coefficients of the macromolecules in GPC has been repeatedly confirmed experimentally [2-4]. It may be supposed that it will also plove adequate on weak adsorption. Because of the approximate nature of formula (2) and the simplification of the polymer and adsorbent models used the proposed method can hardly prove more exact than the former methods of determining polydispersity. However, because of its simplicity it may be useful for express analyses, investigations of new polymers for which standards are absent, for comparative evaluation of samples, etc. if the link between the size and M M of the macromolecules is known the method outlined may also be used to obtain the form of the MD.

Translated by A. CRozY

Chromatographic separation of macromolecules

1025

REFERENCES I. V. N. TSVETKOV, V. Ye. ESKIN and S. Ya. FRENKEL', Struktura makromolekul v rastvorakh (Structure of Macromolecules in Solutions). Moscow, 1964 2. P. P. NEFEDOV and P. N. LAVRENKO, Transportnye metody v analiticheskoi khimii polimerov (Transport Methods in Analytical Polymer Chemistry). Leningrad, 1979 3. B. G. BELEN'KII and L. Z. VILENCHIK, Khromatografiya polimerov (Chromatography of Polymers). Moscow, 1978 4. E. F. CASASSA and Y. TAGAMI, Macromolecules 2: 14, 1969 5. M. B. TENNIKOV and P. P. NEFEDOV, Vysokomol. soyed. A22: 461, 1980 (Translated in Polymer Sci. U.S.S.R. 22: 2, 513, 1980) 6. A. CAMPOS, V. SORIA and J. E. FIGUERUELLO, Macromolec. Chem. 180: 1961, 1979 7. I. B. TSVETKOVSKII, Vysokomol. soyed. A26: 1777, 1984 (Translated in Polymer Sci. U.S.S.R. 26: 8, 1991, 1984) 8. L V. TSVETKOVSKII and R. A. SHLYAKHTER, Zh. analit, khim. 37: 1270, 1982 9. A. A. GORBUNOV and A. M. SKVORTSOV, Vysokomol. soyed. A28: 2453, 1986 (Translated in Polymer Sci. U.S.S.R. 28: I1, 2729, 1986)

Polymer SelencoU.S.S.R. Vol. 29, No. 5, pp. 1025-1031,1987

Printed in Poland

0032-3950[87 $10.00+.00 1988Pergamon Press plc

THEORY OF CHROMATOGRAPHIC SEPARATION OF LINEAR AND RING MACROMOLECULES * A. A. GORBUNOV a n d A. M. SKVORTSOV Leningrad Chemicopharmaceutical Institute

(Received 14 October 1985) A theory is constructed for the chromatographic separation of flexible chain linear and ring macromolecules. Chromatograms are presented calculated for polydisperse ring macromolecules and for a mixture of them with similar linear chains for different energies of adsorption of the units on the pore walls. Analytical expressions have been obtained for the distribution coefficients of the linear and ring molecules and for the effectiveness of their separation in different chromatographic regimes. For the best separation of polydisperse macromolecules by their topology it is necessary to create conditions of weak adsorption intermediate between the regimes of GPC and "critical" chromatography. RING m a c r o m o l e c u l e s p l a y i n g an i m p o r t a n t role in the f u n c t i o n i n g o f living o r g a n i s m s [1, 2] have in t h e last few years b e c o m e one o f t h e test objects in p o l y m e r c h e m i s t r y a n d physics. M e t h o d s have n o w been devised f o r the synthesis o f flexible c h a i n r i n g P D M S [3, 4] a n d PS [5] with M M r e a c h i n g ,,~ l 0 s. The s e p a r a t i o n a n d analysis o f ring * Vysokomol. soyed. A29: No. 5, 926-931, 1987.