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Vibrational spectra of macromolecules with defects
V I B R A T I O N A L SPECTRA OF MACROMOLECULES W I T H D E F E C T S * I. D. MIKHAILOV,V. Z. KOMPANIETS and E. F. OLEINIK I n s t i t u t e of Chemical Physics, U.S.S.R. Academy of Sciences
(Received 6 August 1970) VrSRATIONAL spectroscopy of polymers has now achieved considerable success. Theoretical methods developed in recent years help to estimate t h e frequencies and forms of normal macromolecular vibrations of considerable length, the structure of which satisfy the "geometrical equivalence" of monomer units and polymer crystals formed of these molecules [ 1, 2]. The spectrum of normal vibration frequencies of this t y p e of objects is determined b y the dispersion ratio co~co (0), where 0 is the phase difference of normal vibrations of adjacent monomer units of the polymer chain, an analogue of the lattice wave vector [2]. This fact is due to the presence of a translation s y~nmetry (periodicity) of the polymer object and enables us to factorize dynamic m a t r i x D of the polymer molecule and thus reduce the problem of determining the frequency spectrum of normal vibrations of polymers to solving the secular equation [3-5]
16 (o)-,o,~: I= o,
(11
where ]~ is the unit m a t r i x and ¢o is frequency. I t is quite clear t h a t this approach is only applicable to a comparatively narrow range of polymer specimens, namely high molecular weight crystalline homopolymers since it is in fact only for these t h a t the condition of periodicity of molecular and crystalline structure is fairly satisfactorily fulfilled [6]. The m a j o r i t y of polymer specimens have very " d a m a g e d " structures and different types of disruption of regularity (subsequently termed defects) already exist at molecular level. Among these objects are: statistical eol~lymers, blockcopolymers with different lengths of blocks, graft copolymers and non-crystalline or poorly crystallizing homopolymers, the structural periodicity of which is disrupted as a result of deviations from the normal process of polymerization. The main problem of v i b r a t o r y spectroscopy of polymers is t h e establishment of a relation between the frequency spectrum of normal vibrations of these objects and the number, structure and distribution of various defects in the polymer c h a i n - - h a s up to now therefore remained practically unsolved. To solve this problem the method of Green functions, which is being successfully used for the study of the dynamics of crystalline lattices with defects, offers the most potential possibilities [7-9]. Using this approach, as developed b y m a n y authors, an accurate solution of the problem can be derived [10, 11]. I t is significant t h a t the need for numerical calculation only arises in the last stage of the calculations and normally does not require a large computer memory. I n addition, this method can be used to obtain b o t h the v i b r a t o r y spect r u m of a polymer of any length (with or without translational symmetry) artd an analytical expression for the function of frequency distribution I (co~) which provides the number of frequencies in the interval (co~, o)~q-d (o92), knowledge of which is very i m p o r t a n t in determining the thermodynamic properties of specimens examined [7] and which cannot, in principle, be derived by methods based on the factorization of a dynamic macromolecular matrix. * Vysokomol. soyed. A14: No. 3, 706-713, 1972. 790
Vibrational spectra of macromolecules with defects
791
This paper deals with the use of the method of Green functions to calculate and analyse the vibrational spectra of model polymer chains. Vibrations of linear chains of finite and infinite length and chains with lateral branches simulating graft copolymers were examined. I n every case we will use a harmonic approximation and only take into account the interaction of nearest neighbours. We will be restricted i n t h i s paper to the case in which defects are situated regularly with different concentrations. I t will further be shown how these results m a y be generalized to cover the case of real polymer molecules of helical structure and the result of considering more distant reactions and random arrangement of defects. The Green function of a regular linear chain of infinite length. Let us examine a linear polymer chain of t y p e ( - - A - - ) ~ , where A represents point type masses. We use a formalism which has been described in detail [10] and is based on the Dawson equation for the Green function 0 ~ , ( t ) = - - i 0 (t)<[U~ ( 0 , U~, (0)]>,
0 (t)=
1,
t>O
0,
t<0
(2)
where n and n' are numbers of chain units, U~n is the displacement of the atom in unit n in direction ~ (~, fl=x, y, z), <...> is the average of the equilibrium thermodynamic distribution. I n our approximation the vibratory part of the Hamilton ideal infinite chain takes the form ^
1
2
1
^
Hvtbr=2M- ~n Pn -~ -- Z Unarm' Vn'
(3)
where P~ is the pulse linked with the displacement of Un, M is the mass of atom A, a~V
o'
V is the potential energy and index "0" signifies that the derivative is taken in the equilibrium position. Denoting G°n,(t) dependent on the time of the Green function of an ideal chain, let us present it in the form of the Fourier integral ^
G°,n,(t) = ~
1
o0
^
~ Gnn,(co) e- ~°Cdeo
(4)
Then for the Fourier form of the Green function and bearing in mind the Hamilton expression (3) equation [10] is correct
~o~MG°.,(o~)=Ea..,+C~(2G°~,,(co)- (~° +,,,,,(a~)--d°_ ~,~,.(o~)],
(5)
where E is the unit matrix of the third order, 5... is Kronecker's delta symbol.
I. D. MIKHAII~OV e2 a/.
792
We chose the presentation in which the matrix of force constants ~ is diagonal, i.e. where f , is the force constant, which characterizes the elasticity of the chemical bond A - - A in direction ~. I t was shown [11] that the solution Of equation (b) takes the form G~,(o~)= --[2f, sinh q,]-x, e-q"ln'-hi
(7)
where the following notations were introduced 2o) 2
cosh q~= 1
2
2 =4fdM_l, , o)m~
(7a)
o)max
t
o)m~. is the maximum frequency of the ~ acoustic branch. In equation (7) qffi is generally complex and is situated on the contour indicated in Fig. 1. In the field of resolved frequencies (0 ~ ~ o ) ~ ) q~ is purely imaginary (q~=Imq~) an d in the region of imaginary frequdncies corresponding to disintegration of t h e lattice in the past or the future, real q~=Re q~, while for O)>O)m~ • Im q~=r.
q] lTr
z<-! Izll
g
zF
FIG. 1. Contour of the variation of q. The vibratory spectrum of the system is determined by poles of the Green function [7] which for an ideal linear chain ( - - A - - ) ~ can be found from equation (7), if it is described in a pulse form (i.e. in 6 representation). I n this representation (7) takes the form
~.(o))= -[2A sin.h q~]-~
'~"
cosh q,--cos 0
(8)
For longitudinal vibrations of chain ( - - A - - ) ~ index a which denotes the frequency branches may be omitted since it only contains one acoustic branch. However, results obtained are also valid for helical chains of type (--A--)®, for which there are several (not more than three) acoustic branches therefore index a will further be used everywhere. Introducing the value of W (6) (number of vibrations per unit length of interval 6), which in our case is 1/2g[12], for the function of frequency distribution, considering (8), we obtain /d~\ 1 2
i.
w(6)
(9)
V i b r a t i o n a l speutra of macromolecules w i t h defects
793
The Green function and the vibratory spectrum of colaolymers (--A I-IB--)®. Let us examine the linear model chain of type (--At-IB--)~o, in which A an4 B are point type masses of different values. For simplicity, let us restrict ourselves to the case in which the force constants of bonds A - - A and A - - B are the same (isotopic substitution). It will be shown below how the results obtained are transferred to the case of various forc e constants. C.Omax
(Z I
1
O ~/5
3~/5
~r
FIG. 2. V i b r a t i o n a l s p e c t r u m of a h o m o p o l y m e r ( - - A - - ) ~ (a) a n d a copolymer ( - - A t : I B - - )
(b), where t--~5; l - - e > 0 ;
2--8<0.
The chain of this copolymer can be regarded as a homopolymer of type ( - - A - - ) ~ examined earlier, in which defects of type B appear regularly with concentration c=l/l., i.e. a chain, all the units of which with numbers n ' = 0 , +.ll, +-21, +_31,... contain defects with weights M = M - F A M . The Green function for this chain conforms to the Dawson equation [10] of the type
G.,~, (w)-=G°,,, (w)--w2ZM
~
G°,¢,(w)G,,,,., (m),
(10)
where G=,, 0 @) is the Green function of an ideal chain ( - - A - - ) ~ determined by ratio (7). Bearing in mind that the Fourier type of Green functions separated by distance l in view of the translational s y m m e t r y are the same we substitute in (10) n-=sl(sl=O, +l, +21, ±31 . . . . ) and change to 0 representation. Considering (7) we find
Goo"(w):G°o" (w)--o~2zlM 2---~ 1 f Go"o"(o~) ~ (0, 0")dO",
(11)
where (0, 0 " ) = --[2f~ sinh q, (cosh/q~--cos lO)]-1 sinh lq~-2~6 (0--if') .
(lla)
Substituting further (8) and (lla) in (11) we obtain the Green function of the copolymer chain (--At-xB--)o~ in the form (o~)=cosh/q.--cos lO eosh q~--eos O
'~oo' w2AM sinh lq,, ' eosh/q,+cos lO 2f.
sinh q~
(12)
794
I . D . M ~ A I L o v eta/.
This expression can be described by the Chebyshev polynomials of first and second order (Tl and Uz of I power) as
((a)=Tz(Z~)-cos IO Z~--cos 8
~oo' T~ (Z~)--~ (1 --Z~) U~-I (Z~)--cos 18'
(13)
where e=JM/M, Z~----coshq~and q~, and (amax• have been determined previously. From (13) the vibratory spectrum of the copolymer can be determined, the pole of the Green function being determined from the following ratio:
T~ (Z~)--e(1-Z~) U~-~ (Z~)-cos lO=O (-~
(14)
It is immediately apparent that when l > 1, pole Z=eos/9 which characterizes the spectrum of chain (--A--)~, is eliminated. In special cases l = l , 2, 3 and equation (14) coincides with that derived by other authors [7, 8, 13]. From (13) the function of frequency distribution is immediately derived, which is given by the expression 1
I" (co') dm'=~-~ arc cos [T~ (Z,) --8 (1 --Z,) Uz-1 (Z~)]
(15)
The vibrational spectrum of the copolymer (--A--,_,B--)~ breaks down to l branches, of which the boundary frequencies (when 8 = 0 and O=lr/l) are determined by the ratios
T~(Z~)--~(1-Z~) U,-I(Z~)=I
(O=o) (16)
Tz (Z~)--c (I --Z,) U~-I (Z,)------1
(O=g/l)
A n analysis of equation (14) leads to the important conclusion concerning the type of vibrational spectrum of the copolymer according to the type of defect. If the mass of defect B is larger than mass A (~>0), all I branches in the copolymer spectrum are within the frequency interval 0
~ (I -- ~ ) - ~/2
(I 7)
A further increase in concentration of defects has the result that instead of the local impurity frequency (amp ~ an impurity frequency branch is formed, the width of the frequency interval £2ffiof which depends on the concentration of defects B. Figure 2 shows diagrammatically the results obtained for cases when ~ > 0 (curve 1) and e < 0 (curve 2).
Vibrational spectra of macromolecules with defects
795
I t is indeed very important to know the behaviour of ~ according to the concentration of defects. Defect formation in an ideal chain should result in the disruption of the strict laws of selection, which determine the vibrational spectra of the systems examined [2]. In this case the formation of absorption bands should be expected in the experimentally measured spectra with frequencies formerly prohibited by rules of selection, the intensities of these being proportional to the number of frequencies in a given frequency range. All this should result in a deformation of absorption bands observed in the vibrational spectrum of the copolymer, whereby it could be expected that a widening and asymmetry of corresponding absorption bands will be proportional to variations in the value of 1ft. For low concentrations of defects (l>>l) a dispersion curve co~----co~(0) can be obtained for the impurity frequency branch, solving equations (12) and (13) by a method of successive approximations
It follows from (18) that the dependence of ~" on concentration c of defects is of the non-analytical type ~ 4~ ~ [ 1 + ~ z
,<0.1>>1)
(19)
In addition, with an increase in defect concentration, the centre of the frequency range i9~ is displaced in the direction of lower frequencies by a value A'--
-~"
2~A ( 1 + " ~ 'z (1 -s~) '~, \i-17e/
(20)
The latter result is of particular practical importance as it predicts the displacement of the centre of absorption band (the rule of selection being disrupted) to the direction of lower frequencies. For the impurity branch from (14) the function of frequency distribution can be found, which takes the form i~(o)2)do)2=_~l arc cos [ ( l + t ' ~ '
kk
\T-$71
\T-$VIj
(21)
The Green function and the vibrational ,pectrum of chains of finite length. Considerable attention is being given in the literature to the problem of vibrations of chains of finite length [14, 15]. The method used in former chapters may be applied to solve this problem. We examine the following model. In a linear ideal chain of infinite length in units with numbers sl and s / + l (s=0, =hi, + 2 , . . . )
I. D.
796
Mr~A,T.OV ~
a/.
we place defects with masses MI-~M+AM, between which the interaction is described b y force constantf'~----f~-Af~, where f~ is the elasticity of the chemical bond A - - A in direction a (a~x, y, z). I f in this model f~l = 0 , the initial chain breaks down to many chains of length l, the end groups of which differ from the groups of the m a i n chain. Without pausing to solve the Dawson equation for this model, which is described in another study [16], we only cite the final results. The vibrational spectrum of this model is determined from the equation Ti(Z~)--(I~7~) cos/0--(2~--?~--2aZ~) U~-I (Z~)
+[28~(1--Za)2--28 (1 --Z~) ~--y.] Ut-2(Z~)=0 ;
(22)
Assuming that y a = - - I (fa~=0), we find TI(Z~)--[I~-2s (1--Z~)] Ul_l(Z~)-F[282(1-Z~) ~
(23)
+2~ ( I ~ - Z , ) + I ] Uz_2(Z~)=O Equation (23) is independent of 0 and is an equation of lth order in relation to Z~ and has therefore 1 discrete'solUtions. All these roots are within the frequency interval 0
(eq'--r~)~--e-2tq~(1--r~eq')2----O ,
(23a)
where 2o)2 cosh q~--~l----V--- < - - 1 , (L) m a x
r~= 1A-2e ( I + Z ~ ) < O
(23b)
tt
Solving (23a) b y successive approximation for l>>l we find that when e<21there are two solutions ~±--
¢Omaxa 4~ ( e + l )
1-4-
]
(2~-+-1)2 ,
( - - 1 < ~ < _1/~)
(24)
Normal vibration frequencies which are higher than frequency O)max~ are only formed when the mass of terminal chain groups is such that the condition Mconc<2M holds good. Furthermore, it follows from (24) that the distance between these frequencies rapidly decreases with chain length; the larger the distance between ½-Mconc/M, the more rapid is this reduction. Vibrational spectrum of linear chain8 with side branches. The problem concerning the vibrational spectrum of chains with side chains can be reduced to the problem previously solved for the copolymer (--A t_lB--) ~. For simplicity we examine a model chain (--A z_IABr--)~, in which a side chain is added to each unit with a number sl (s----0, ± 1 , ± 2 . . . . ); the lengths of all side chains are the
Vibrational spectra of macromolecules with defects
797
same and equal to r. Results of calculations are readily generalized for the case of several side chains of different length. Let us introduce the following notations: r is the length (number of a t o m s ) o f the side chain; M is the mass of atoms of the main "skeletal" chain; M~ is the mass of atoms of the side chain; ~ is the matrix of force constants of the "skeletal" chain and ~1 is the matrix of force constants of the side chain. We note that matrices ~ and ~1 generally speaking do not commutate. The Dawson equation for this chain takes the form
(~M~-2~,)0x,,.. ( ~ ) = ~ 6 . . . - ~ [Ox.+l,,,,(o~)+ d.-l, .,(o~)] + ~1[dnn,(~) -- ~,nn,(~o)] (~MI~ -- 2+,) ~,,,.,(io)=- ~.[d,n-1, n,('O)+d,n+1, n,(~)] (m=:,~ ..... ,'; do, n,(~)=~nn,(~), d,+., n,(~) =O)
(25)
Here indices n and n' refer to the main chain and m to the side chain. Solving the system of last r of equations (25) in relation to G~n, @) we find that
~n'(~)=--h;1(~o) (LICc~) ~nn,(~) , where A-k(co)=(o93M~E-~l) Uk-1
(26)
(Zl)--~lV/f-,(Zl)
(26a)
1
2 Considering (26), the first equation in (25) may be rewritten as
( ~ ¢ . - ~ ) ~nn,(~O)=~Snn,--~[~n., n'(~)+ ~,,-I, n,(~O)] --~V1~g~)~,,n,(~),
(27)
where An analysis of these expressions indicates that the presence of a side chain at the unit with number n is equivalent to the mass variation of this u n i t b y a value AM (e)), which is determined b y ratios (27a) and (26a). The problem of determining the spectrum of this chain is thus reduced to finding the spectrum of copolymer (--A ~-~B--), for which MB=MA--~-AM (~o). Therefore, using the results obtained and bearing in mind that AI~I (w) m a y not commutate with ~, we have the equation
[Tr ( ~ ) - ~ cos r0] [(~2M~_ $1) Vr-1 (Zl) - $1U~-2 (~1)] -1125",_i (~)[(~o~Mi~-,~i) U~-1 (~I)+(~MI~--~I) U,_,(~,) -~lur-,(~,)]=0 1
2
^
'
1
2
which determines the spectrum of this model.
(2s) .,
,, '
798
I.D. Mr~Lov
eta/.
Finally, we n o t e t h a t all t h e results can be easily applied to a n o t h e r t y p e of defect, n a m e l y w h e n the matrices of force constants (case of conformation defects in t h e polymer) a n d n o t the masses change and also t o t h e most general case, w h e n b o t h change. F o r this, the general disturbance due to defects should be simply considered, which is given b y the expression [10].
A V~,(w)-~ --eo2z~MnJnn,J~Pq-A~,
(29)
CONCLUSIONS
(1) A s t u d y was m a d e of the application o f the Green function for t h e analysis o f vibrational spectra o f polymers and copolymers using several models: a linear h o m o p o l y m e r of infinite length, a linear c o p o l y m e r o f infinite length, a linear h o m o p o l y m e r of finite length and a linear h o m o p o l y m e r with side chains (graft copolymer). (2) An a n a l y t i c a l expression was derived for t h e Green function o f the above models. (3) An analytical expression was o b t a i n e d for the dispersion and f r e q u e n c y distribution function and the properties of the s p e c t r u m analysed according to t h e c o n c e n t r a t i o n and t y p e o f defect.
Translated by E. S~XERE REFERENCES
1. Novoishie metody issledovaniya polimerov (Latest Methods of Investigating Polymers) B. KI, (Ed.), Izd. "Mir", 1966 2. E. F. OLEINIK and g. Z. KOMPANIETS, Uspekhi v oblasti kolobatel'noi spoktroskopii polimerov v sb. Novoe v metodakh issledovaniya polimerov (Progress in the Field of Vibrational Spectroscopy of Polymers. Syrup.: Latest Methods of Studying Polymers). Izd. "Mir", 1968 3. E. WILSON, G. DESIUS and P. CROSS, Tooriya kolobatel'nykh spektrov molekul (Theory of Vibrational Spectra of Molecules). Izd. inostr, lit., 1960 4. L. S. MAYANTS, Teoriya i raschot kolebanii molekul (Theory axed Calculation of Molecular Vibrations). Izd. inostr, lit., 1960 5. L. A. GRIBOV, Vvedonie v teoriyu i raschet kolobatel'nykh spektrov mnogoatonmykh molekul (Introduction into the Theory and Calculation related to Vibrational Spectra of Polyatomic Molecules). Izd. LGU, 1965 6. B. K. VAINSHTEIN, Diffraktsiya rentgcnovskikh luchei na tsopnykh molekulakh (X-ray DiffTaction on Chain Molecules). Izd. AN SSSR, 1963 7. A. MARADUDIN, E. MONTROLL and G. WEISS, Dynamicheskaya teoriya kristallicheskoi reshetki v garmonicheskom priblizhenii (Dynamic Crystalline Lattice Theory in Harmonic Approximation). Izd. "Mix", 1965 8. A. MARADUDIN, Defekty i kolcbatel'nyi spektr kristallov (Defects and Vibrational Spectrum of Crystals). Izd. "Mir", 1968 9. I. M. LIFSHITS, Uspekhi fiz. n. 83: 617, 1964 10. Yu. KAGAN, Materialy shkoly pc teorii defektov v kristallakh (Data of the School on the Theory of Defects in Crystals). Tbilisi, 1966, 11. S. I. KUBAREV and I. D. MII~HAILOV, Fizika met. i metalloved. 27: 29, 1969
Study of molecular weight and composition polydispersion of copolymers
799
12. Oh. KITTEL', Vvedenie v fiziku tverdogo tela (Introduction into Solid State Physics). Fizmatgiz, 1957 13. A. MARADUDIN and G. WEISS, J. Chem. Phys. 29, 631, 1959 14. I. lAKES, Collect. czechoslov, chem. Communic. 5: 1523, 1965 15. L. A. GRIROV and T. S. ABILOVA, Optika i spektroskopiya 23, 374, 535, 1967 16. S. L KUBAREV and I. D. MIKHAIL()V, Zh. teoret, i eksper, khimii 5: 646, 1969
STUDY OF THE MOLECULAR WEIGHT AND COMPOSITION POLYDISPERSION OF COPOLYMERS BY RAPID SEDIMENTATION* I. YA. PODDUBNYI,A. V. PODALINSKIIand V. A. GRECHAI~OVSKII The S. V. Lebedev All-Union Scientific Research Institute for Synthetic Rubber
(Received 21 September 1970) To STUDY the structural properties of copolymers, particularly heterogeneity of composition several physical and chemical methods have been recently developed [1-3]. However, the possibilities offered by these methods are still in m a n y ways limited, therefore the problem of developing new methods to characterize copolymer chain structure continues to be urgent. The method proposed is based on the simultaneous study by sedimentation experiments of two independent optical parameters of the copolymer studied: the refractive index and the index of absorption of ultraviolet radiation of a certain wave length. Characteristics of heterogeneity. Prior to describing the method we introduce the functions subsequently used, by which the heterogeneity of copolymers is characterized according to molecular weight M and composition. The first function qw (M) is a differential gravimetric function of molecular weight distribution (MWD); the product of qw (M)AM characterizes the weight fraction of the polymer, the molecular weight of which is in the range of M to M-~AM. The second function ~ (M) stands for the dependence of average composition of macromolecules with a given M on molecular weight
~=(x (M )=Wl/W ,
(1)
where wl is the weight of component 1 in the macromolecules with molecular weight ranging from M to M q - A M , w being the overall weight of polymer molecules in the range of molecular weights M , M q - A M . This function describes the composition in fractions of different molecular weight irrespective of their weight content in the polymer. Therefore, for a more complete description of the composi* Vysokomol. soyed. AI4: :No. 3, 714-721, 1972.
(Received 6 August 1970) VrSRATIONAL spectroscopy of polymers has now achieved considerable success. Theoretical methods developed in recent years help to estimate t h e frequencies and forms of normal macromolecular vibrations of considerable length, the structure of which satisfy the "geometrical equivalence" of monomer units and polymer crystals formed of these molecules [ 1, 2]. The spectrum of normal vibration frequencies of this t y p e of objects is determined b y the dispersion ratio co~co (0), where 0 is the phase difference of normal vibrations of adjacent monomer units of the polymer chain, an analogue of the lattice wave vector [2]. This fact is due to the presence of a translation s y~nmetry (periodicity) of the polymer object and enables us to factorize dynamic m a t r i x D of the polymer molecule and thus reduce the problem of determining the frequency spectrum of normal vibrations of polymers to solving the secular equation [3-5]
16 (o)-,o,~: I= o,
(11
where ]~ is the unit m a t r i x and ¢o is frequency. I t is quite clear t h a t this approach is only applicable to a comparatively narrow range of polymer specimens, namely high molecular weight crystalline homopolymers since it is in fact only for these t h a t the condition of periodicity of molecular and crystalline structure is fairly satisfactorily fulfilled [6]. The m a j o r i t y of polymer specimens have very " d a m a g e d " structures and different types of disruption of regularity (subsequently termed defects) already exist at molecular level. Among these objects are: statistical eol~lymers, blockcopolymers with different lengths of blocks, graft copolymers and non-crystalline or poorly crystallizing homopolymers, the structural periodicity of which is disrupted as a result of deviations from the normal process of polymerization. The main problem of v i b r a t o r y spectroscopy of polymers is t h e establishment of a relation between the frequency spectrum of normal vibrations of these objects and the number, structure and distribution of various defects in the polymer c h a i n - - h a s up to now therefore remained practically unsolved. To solve this problem the method of Green functions, which is being successfully used for the study of the dynamics of crystalline lattices with defects, offers the most potential possibilities [7-9]. Using this approach, as developed b y m a n y authors, an accurate solution of the problem can be derived [10, 11]. I t is significant t h a t the need for numerical calculation only arises in the last stage of the calculations and normally does not require a large computer memory. I n addition, this method can be used to obtain b o t h the v i b r a t o r y spect r u m of a polymer of any length (with or without translational symmetry) artd an analytical expression for the function of frequency distribution I (co~) which provides the number of frequencies in the interval (co~, o)~q-d (o92), knowledge of which is very i m p o r t a n t in determining the thermodynamic properties of specimens examined [7] and which cannot, in principle, be derived by methods based on the factorization of a dynamic macromolecular matrix. * Vysokomol. soyed. A14: No. 3, 706-713, 1972. 790
Vibrational spectra of macromolecules with defects
791
This paper deals with the use of the method of Green functions to calculate and analyse the vibrational spectra of model polymer chains. Vibrations of linear chains of finite and infinite length and chains with lateral branches simulating graft copolymers were examined. I n every case we will use a harmonic approximation and only take into account the interaction of nearest neighbours. We will be restricted i n t h i s paper to the case in which defects are situated regularly with different concentrations. I t will further be shown how these results m a y be generalized to cover the case of real polymer molecules of helical structure and the result of considering more distant reactions and random arrangement of defects. The Green function of a regular linear chain of infinite length. Let us examine a linear polymer chain of t y p e ( - - A - - ) ~ , where A represents point type masses. We use a formalism which has been described in detail [10] and is based on the Dawson equation for the Green function 0 ~ , ( t ) = - - i 0 (t)<[U~ ( 0 , U~, (0)]>,
0 (t)=
1,
t>O
0,
t<0
(2)
where n and n' are numbers of chain units, U~n is the displacement of the atom in unit n in direction ~ (~, fl=x, y, z), <...> is the average of the equilibrium thermodynamic distribution. I n our approximation the vibratory part of the Hamilton ideal infinite chain takes the form ^
1
2
1
^
Hvtbr=2M- ~n Pn -~ -- Z Unarm' Vn'
(3)
where P~ is the pulse linked with the displacement of Un, M is the mass of atom A, a~V
o'
V is the potential energy and index "0" signifies that the derivative is taken in the equilibrium position. Denoting G°n,(t) dependent on the time of the Green function of an ideal chain, let us present it in the form of the Fourier integral ^
G°,n,(t) = ~
1
o0
^
~ Gnn,(co) e- ~°Cdeo
(4)
Then for the Fourier form of the Green function and bearing in mind the Hamilton expression (3) equation [10] is correct
~o~MG°.,(o~)=Ea..,+C~(2G°~,,(co)- (~° +,,,,,(a~)--d°_ ~,~,.(o~)],
(5)
where E is the unit matrix of the third order, 5... is Kronecker's delta symbol.
I. D. MIKHAII~OV e2 a/.
792
We chose the presentation in which the matrix of force constants ~ is diagonal, i.e. where f , is the force constant, which characterizes the elasticity of the chemical bond A - - A in direction ~. I t was shown [11] that the solution Of equation (b) takes the form G~,(o~)= --[2f, sinh q,]-x, e-q"ln'-hi
(7)
where the following notations were introduced 2o) 2
cosh q~= 1
2
2 =4fdM_l, , o)m~
(7a)
o)max
t
o)m~. is the maximum frequency of the ~ acoustic branch. In equation (7) qffi is generally complex and is situated on the contour indicated in Fig. 1. In the field of resolved frequencies (0 ~ ~ o ) ~ ) q~ is purely imaginary (q~=Imq~) an d in the region of imaginary frequdncies corresponding to disintegration of t h e lattice in the past or the future, real q~=Re q~, while for O)>O)m~ • Im q~=r.
q] lTr
z<-! Izll
g
zF
FIG. 1. Contour of the variation of q. The vibratory spectrum of the system is determined by poles of the Green function [7] which for an ideal linear chain ( - - A - - ) ~ can be found from equation (7), if it is described in a pulse form (i.e. in 6 representation). I n this representation (7) takes the form
~.(o))= -[2A sin.h q~]-~
'~"
cosh q,--cos 0
(8)
For longitudinal vibrations of chain ( - - A - - ) ~ index a which denotes the frequency branches may be omitted since it only contains one acoustic branch. However, results obtained are also valid for helical chains of type (--A--)®, for which there are several (not more than three) acoustic branches therefore index a will further be used everywhere. Introducing the value of W (6) (number of vibrations per unit length of interval 6), which in our case is 1/2g[12], for the function of frequency distribution, considering (8), we obtain /d~\ 1 2
i.
w(6)
(9)
V i b r a t i o n a l speutra of macromolecules w i t h defects
793
The Green function and the vibratory spectrum of colaolymers (--A I-IB--)®. Let us examine the linear model chain of type (--At-IB--)~o, in which A an4 B are point type masses of different values. For simplicity, let us restrict ourselves to the case in which the force constants of bonds A - - A and A - - B are the same (isotopic substitution). It will be shown below how the results obtained are transferred to the case of various forc e constants. C.Omax
(Z I
1
O ~/5
3~/5
~r
FIG. 2. V i b r a t i o n a l s p e c t r u m of a h o m o p o l y m e r ( - - A - - ) ~ (a) a n d a copolymer ( - - A t : I B - - )
(b), where t--~5; l - - e > 0 ;
2--8<0.
The chain of this copolymer can be regarded as a homopolymer of type ( - - A - - ) ~ examined earlier, in which defects of type B appear regularly with concentration c=l/l., i.e. a chain, all the units of which with numbers n ' = 0 , +.ll, +-21, +_31,... contain defects with weights M = M - F A M . The Green function for this chain conforms to the Dawson equation [10] of the type
G.,~, (w)-=G°,,, (w)--w2ZM
~
G°,¢,(w)G,,,,., (m),
(10)
where G=,, 0 @) is the Green function of an ideal chain ( - - A - - ) ~ determined by ratio (7). Bearing in mind that the Fourier type of Green functions separated by distance l in view of the translational s y m m e t r y are the same we substitute in (10) n-=sl(sl=O, +l, +21, ±31 . . . . ) and change to 0 representation. Considering (7) we find
Goo"(w):G°o" (w)--o~2zlM 2---~ 1 f Go"o"(o~) ~ (0, 0")dO",
(11)
where (0, 0 " ) = --[2f~ sinh q, (cosh/q~--cos lO)]-1 sinh lq~-2~6 (0--if') .
(lla)
Substituting further (8) and (lla) in (11) we obtain the Green function of the copolymer chain (--At-xB--)o~ in the form (o~)=cosh/q.--cos lO eosh q~--eos O
'~oo' w2AM sinh lq,, ' eosh/q,+cos lO 2f.
sinh q~
(12)
794
I . D . M ~ A I L o v eta/.
This expression can be described by the Chebyshev polynomials of first and second order (Tl and Uz of I power) as
((a)=Tz(Z~)-cos IO Z~--cos 8
~oo' T~ (Z~)--~ (1 --Z~) U~-I (Z~)--cos 18'
(13)
where e=JM/M, Z~----coshq~and q~, and (amax• have been determined previously. From (13) the vibratory spectrum of the copolymer can be determined, the pole of the Green function being determined from the following ratio:
T~ (Z~)--e(1-Z~) U~-~ (Z~)-cos lO=O (-~
(14)
It is immediately apparent that when l > 1, pole Z=eos/9 which characterizes the spectrum of chain (--A--)~, is eliminated. In special cases l = l , 2, 3 and equation (14) coincides with that derived by other authors [7, 8, 13]. From (13) the function of frequency distribution is immediately derived, which is given by the expression 1
I" (co') dm'=~-~ arc cos [T~ (Z,) --8 (1 --Z,) Uz-1 (Z~)]
(15)
The vibrational spectrum of the copolymer (--A--,_,B--)~ breaks down to l branches, of which the boundary frequencies (when 8 = 0 and O=lr/l) are determined by the ratios
T~(Z~)--~(1-Z~) U,-I(Z~)=I
(O=o) (16)
Tz (Z~)--c (I --Z,) U~-I (Z,)------1
(O=g/l)
A n analysis of equation (14) leads to the important conclusion concerning the type of vibrational spectrum of the copolymer according to the type of defect. If the mass of defect B is larger than mass A (~>0), all I branches in the copolymer spectrum are within the frequency interval 0
~ (I -- ~ ) - ~/2
(I 7)
A further increase in concentration of defects has the result that instead of the local impurity frequency (amp ~ an impurity frequency branch is formed, the width of the frequency interval £2ffiof which depends on the concentration of defects B. Figure 2 shows diagrammatically the results obtained for cases when ~ > 0 (curve 1) and e < 0 (curve 2).
Vibrational spectra of macromolecules with defects
795
I t is indeed very important to know the behaviour of ~ according to the concentration of defects. Defect formation in an ideal chain should result in the disruption of the strict laws of selection, which determine the vibrational spectra of the systems examined [2]. In this case the formation of absorption bands should be expected in the experimentally measured spectra with frequencies formerly prohibited by rules of selection, the intensities of these being proportional to the number of frequencies in a given frequency range. All this should result in a deformation of absorption bands observed in the vibrational spectrum of the copolymer, whereby it could be expected that a widening and asymmetry of corresponding absorption bands will be proportional to variations in the value of 1ft. For low concentrations of defects (l>>l) a dispersion curve co~----co~(0) can be obtained for the impurity frequency branch, solving equations (12) and (13) by a method of successive approximations
It follows from (18) that the dependence of ~" on concentration c of defects is of the non-analytical type ~ 4~ ~ [ 1 + ~ z
,<0.1>>1)
(19)
In addition, with an increase in defect concentration, the centre of the frequency range i9~ is displaced in the direction of lower frequencies by a value A'--
-~"
2~A ( 1 + " ~ 'z (1 -s~) '~, \i-17e/
(20)
The latter result is of particular practical importance as it predicts the displacement of the centre of absorption band (the rule of selection being disrupted) to the direction of lower frequencies. For the impurity branch from (14) the function of frequency distribution can be found, which takes the form i~(o)2)do)2=_~l arc cos [ ( l + t ' ~ '
kk
\T-$71
\T-$VIj
(21)
The Green function and the vibrational ,pectrum of chains of finite length. Considerable attention is being given in the literature to the problem of vibrations of chains of finite length [14, 15]. The method used in former chapters may be applied to solve this problem. We examine the following model. In a linear ideal chain of infinite length in units with numbers sl and s / + l (s=0, =hi, + 2 , . . . )
I. D.
796
Mr~A,T.OV ~
a/.
we place defects with masses MI-~M+AM, between which the interaction is described b y force constantf'~----f~-Af~, where f~ is the elasticity of the chemical bond A - - A in direction a (a~x, y, z). I f in this model f~l = 0 , the initial chain breaks down to many chains of length l, the end groups of which differ from the groups of the m a i n chain. Without pausing to solve the Dawson equation for this model, which is described in another study [16], we only cite the final results. The vibrational spectrum of this model is determined from the equation Ti(Z~)--(I~7~) cos/0--(2~--?~--2aZ~) U~-I (Z~)
+[28~(1--Za)2--28 (1 --Z~) ~--y.] Ut-2(Z~)=0 ;
(22)
Assuming that y a = - - I (fa~=0), we find TI(Z~)--[I~-2s (1--Z~)] Ul_l(Z~)-F[282(1-Z~) ~
(23)
+2~ ( I ~ - Z , ) + I ] Uz_2(Z~)=O Equation (23) is independent of 0 and is an equation of lth order in relation to Z~ and has therefore 1 discrete'solUtions. All these roots are within the frequency interval 0
(eq'--r~)~--e-2tq~(1--r~eq')2----O ,
(23a)
where 2o)2 cosh q~--~l----V--- < - - 1 , (L) m a x
r~= 1A-2e ( I + Z ~ ) < O
(23b)
tt
Solving (23a) b y successive approximation for l>>l we find that when e<21there are two solutions ~±--
¢Omaxa 4~ ( e + l )
1-4-
]
(2~-+-1)2 ,
( - - 1 < ~ < _1/~)
(24)
Normal vibration frequencies which are higher than frequency O)max~ are only formed when the mass of terminal chain groups is such that the condition Mconc<2M holds good. Furthermore, it follows from (24) that the distance between these frequencies rapidly decreases with chain length; the larger the distance between ½-Mconc/M, the more rapid is this reduction. Vibrational spectrum of linear chain8 with side branches. The problem concerning the vibrational spectrum of chains with side chains can be reduced to the problem previously solved for the copolymer (--A t_lB--) ~. For simplicity we examine a model chain (--A z_IABr--)~, in which a side chain is added to each unit with a number sl (s----0, ± 1 , ± 2 . . . . ); the lengths of all side chains are the
Vibrational spectra of macromolecules with defects
797
same and equal to r. Results of calculations are readily generalized for the case of several side chains of different length. Let us introduce the following notations: r is the length (number of a t o m s ) o f the side chain; M is the mass of atoms of the main "skeletal" chain; M~ is the mass of atoms of the side chain; ~ is the matrix of force constants of the "skeletal" chain and ~1 is the matrix of force constants of the side chain. We note that matrices ~ and ~1 generally speaking do not commutate. The Dawson equation for this chain takes the form
(~M~-2~,)0x,,.. ( ~ ) = ~ 6 . . . - ~ [Ox.+l,,,,(o~)+ d.-l, .,(o~)] + ~1[dnn,(~) -- ~,nn,(~o)] (~MI~ -- 2+,) ~,,,.,(io)=- ~.[d,n-1, n,('O)+d,n+1, n,(~)] (m=:,~ ..... ,'; do, n,(~)=~nn,(~), d,+., n,(~) =O)
(25)
Here indices n and n' refer to the main chain and m to the side chain. Solving the system of last r of equations (25) in relation to G~n, @) we find that
~n'(~)=--h;1(~o) (LICc~) ~nn,(~) , where A-k(co)=(o93M~E-~l) Uk-1
(26)
(Zl)--~lV/f-,(Zl)
(26a)
1
2 Considering (26), the first equation in (25) may be rewritten as
( ~ ¢ . - ~ ) ~nn,(~O)=~Snn,--~[~n., n'(~)+ ~,,-I, n,(~O)] --~V1~g~)~,,n,(~),
(27)
where An analysis of these expressions indicates that the presence of a side chain at the unit with number n is equivalent to the mass variation of this u n i t b y a value AM (e)), which is determined b y ratios (27a) and (26a). The problem of determining the spectrum of this chain is thus reduced to finding the spectrum of copolymer (--A ~-~B--), for which MB=MA--~-AM (~o). Therefore, using the results obtained and bearing in mind that AI~I (w) m a y not commutate with ~, we have the equation
[Tr ( ~ ) - ~ cos r0] [(~2M~_ $1) Vr-1 (Zl) - $1U~-2 (~1)] -1125",_i (~)[(~o~Mi~-,~i) U~-1 (~I)+(~MI~--~I) U,_,(~,) -~lur-,(~,)]=0 1
2
^
'
1
2
which determines the spectrum of this model.
(2s) .,
,, '
798
I.D. Mr~Lov
eta/.
Finally, we n o t e t h a t all t h e results can be easily applied to a n o t h e r t y p e of defect, n a m e l y w h e n the matrices of force constants (case of conformation defects in t h e polymer) a n d n o t the masses change and also t o t h e most general case, w h e n b o t h change. F o r this, the general disturbance due to defects should be simply considered, which is given b y the expression [10].
A V~,(w)-~ --eo2z~MnJnn,J~Pq-A~,
(29)
CONCLUSIONS
(1) A s t u d y was m a d e of the application o f the Green function for t h e analysis o f vibrational spectra o f polymers and copolymers using several models: a linear h o m o p o l y m e r of infinite length, a linear c o p o l y m e r o f infinite length, a linear h o m o p o l y m e r of finite length and a linear h o m o p o l y m e r with side chains (graft copolymer). (2) An a n a l y t i c a l expression was derived for t h e Green function o f the above models. (3) An analytical expression was o b t a i n e d for the dispersion and f r e q u e n c y distribution function and the properties of the s p e c t r u m analysed according to t h e c o n c e n t r a t i o n and t y p e o f defect.
Translated by E. S~XERE REFERENCES
1. Novoishie metody issledovaniya polimerov (Latest Methods of Investigating Polymers) B. KI, (Ed.), Izd. "Mir", 1966 2. E. F. OLEINIK and g. Z. KOMPANIETS, Uspekhi v oblasti kolobatel'noi spoktroskopii polimerov v sb. Novoe v metodakh issledovaniya polimerov (Progress in the Field of Vibrational Spectroscopy of Polymers. Syrup.: Latest Methods of Studying Polymers). Izd. "Mir", 1968 3. E. WILSON, G. DESIUS and P. CROSS, Tooriya kolobatel'nykh spektrov molekul (Theory of Vibrational Spectra of Molecules). Izd. inostr, lit., 1960 4. L. S. MAYANTS, Teoriya i raschot kolebanii molekul (Theory axed Calculation of Molecular Vibrations). Izd. inostr, lit., 1960 5. L. A. GRIBOV, Vvedonie v teoriyu i raschet kolobatel'nykh spektrov mnogoatonmykh molekul (Introduction into the Theory and Calculation related to Vibrational Spectra of Polyatomic Molecules). Izd. LGU, 1965 6. B. K. VAINSHTEIN, Diffraktsiya rentgcnovskikh luchei na tsopnykh molekulakh (X-ray DiffTaction on Chain Molecules). Izd. AN SSSR, 1963 7. A. MARADUDIN, E. MONTROLL and G. WEISS, Dynamicheskaya teoriya kristallicheskoi reshetki v garmonicheskom priblizhenii (Dynamic Crystalline Lattice Theory in Harmonic Approximation). Izd. "Mix", 1965 8. A. MARADUDIN, Defekty i kolcbatel'nyi spektr kristallov (Defects and Vibrational Spectrum of Crystals). Izd. "Mir", 1968 9. I. M. LIFSHITS, Uspekhi fiz. n. 83: 617, 1964 10. Yu. KAGAN, Materialy shkoly pc teorii defektov v kristallakh (Data of the School on the Theory of Defects in Crystals). Tbilisi, 1966, 11. S. I. KUBAREV and I. D. MII~HAILOV, Fizika met. i metalloved. 27: 29, 1969
Study of molecular weight and composition polydispersion of copolymers
799
12. Oh. KITTEL', Vvedenie v fiziku tverdogo tela (Introduction into Solid State Physics). Fizmatgiz, 1957 13. A. MARADUDIN and G. WEISS, J. Chem. Phys. 29, 631, 1959 14. I. lAKES, Collect. czechoslov, chem. Communic. 5: 1523, 1965 15. L. A. GRIROV and T. S. ABILOVA, Optika i spektroskopiya 23, 374, 535, 1967 16. S. L KUBAREV and I. D. MIKHAIL()V, Zh. teoret, i eksper, khimii 5: 646, 1969
STUDY OF THE MOLECULAR WEIGHT AND COMPOSITION POLYDISPERSION OF COPOLYMERS BY RAPID SEDIMENTATION* I. YA. PODDUBNYI,A. V. PODALINSKIIand V. A. GRECHAI~OVSKII The S. V. Lebedev All-Union Scientific Research Institute for Synthetic Rubber
(Received 21 September 1970) To STUDY the structural properties of copolymers, particularly heterogeneity of composition several physical and chemical methods have been recently developed [1-3]. However, the possibilities offered by these methods are still in m a n y ways limited, therefore the problem of developing new methods to characterize copolymer chain structure continues to be urgent. The method proposed is based on the simultaneous study by sedimentation experiments of two independent optical parameters of the copolymer studied: the refractive index and the index of absorption of ultraviolet radiation of a certain wave length. Characteristics of heterogeneity. Prior to describing the method we introduce the functions subsequently used, by which the heterogeneity of copolymers is characterized according to molecular weight M and composition. The first function qw (M) is a differential gravimetric function of molecular weight distribution (MWD); the product of qw (M)AM characterizes the weight fraction of the polymer, the molecular weight of which is in the range of M to M-~AM. The second function ~ (M) stands for the dependence of average composition of macromolecules with a given M on molecular weight
~=(x (M )=Wl/W ,
(1)
where wl is the weight of component 1 in the macromolecules with molecular weight ranging from M to M q - A M , w being the overall weight of polymer molecules in the range of molecular weights M , M q - A M . This function describes the composition in fractions of different molecular weight irrespective of their weight content in the polymer. Therefore, for a more complete description of the composi* Vysokomol. soyed. AI4: :No. 3, 714-721, 1972.