The phase diagram of a macromolecule containing mesogenic groups

~Polymer Science U.S.S.R. Vol. 22, pp. 112-118. O Pergamon Press Ltd. 1980. Printed in Poland

0082--3950/80/0101-0112507.50/0

THE, PHASE DIAGRAM OF A MACROMOLECULE CONTAINING MESOGENIC GROUPS* A. Yu. GROSBERG Chemical Physics Institute, U.S.~R. Academy of Scmnces (Received 23 October 1979) A m o d e l of a m a e r o m o l e c u l e i n c o r p o r a t i n g r o d - h k e m e s o g e n i c g r o u p s i n a flexible n o n - c o r p o r a l c h a i n e i t h e r as b r a n c h e s or as p a r t s of t h e m a i n c h a i n is e x a m i n e d . T h e p r o p e r t i e s of b o t h t h e s e m o d e l s h a v i n g l o n g c h a i n s will b e i d e n t m a l w h e n t h e a p p r o p r i a t e p a r a m e t e r s are selected. T h e c o i l - g l o b u l e t r a n s i t m n w~ll t a k e place i n l o n g c h a i n s a b o v e t h e 0 - p o i n t a n d m of first o r d e r as t o p h a s e ; i t m a c c o m p a n i e d b y a n o r m n t a t l o n a l r e a r r a n g e m e n t of t h e rods inside t h e globule. T h e m e t a s t a b l e , l s o t r o p m s t a t e is also e x a m i n e d a n d a r a n g e f o u n d i n w h i c h a n i s o t r o p i e globule is h k e l y t o exist.

THE study of macromolecules containing groupings of low molecular weight which are able to form liquid crystalline mesophases is important from the viewpoint of the variety of appendages which can be present. A review of the available experimental material can be fonnd in the book by Papkov [1] and in the paper b y Shibayev and Plate [2]. The use of these results is made difficult owing to t h e lack of any. theoretical support for such systems. It was only recently t h a t . some phase diagrams were published [3-5] for a macromolecular solution of rigid [3] and semi-rigid chains in isotropie [4] and anisotropic [5] states. The first step towards a formulation of a theory valid for various chain types was taken by Lifshits [6-8] in his article [9] dealing with a single, isolated macromolecule. An investigation of this kind is of fundamental interest as well as for its wider applications, primary the biological ones. The equations published in an ea~rlier communication [9] were formulated for relatively short molecules with mesogenic groups in the main chain [9] or in branches [10]. We shall examine here the changes in the steric structures and in the thermodynamic properties when such short molecules are extended. A whole series of new and qualitative features, of which some show little sensitivity to • he selection of the model, will be discussed. One must remember when longer chains are examined that specifically long chains will behave universally as i n an ideal solvent (this was clearly shown for the first time by De Gennes [11]; • see also reviews [7, 8], and behaviour in the 0-range [7, 8, 12, 13]). As to the metho~lical aspects, it is worthwhile to examine on a concrete chain model [4] the way ~in which the universal properties form. * V y s o k o m o l . soyod. A22: No. 1, 100-104, 1980. 112

Macromolecule contamlng mesoger~c groups

113

As the chain can have a particular steric structure only in the globular state [6], we shall concentrate our attention on the latter, and also on the analysis of the globule-coil transition. We shall just enumerate the main definitions used and the formulae which had been published before [9]. A chain composed of units of solid rods is under examination, the rods are modelled by mesogenic groups. The length of the rod, l, exceeds its thickness, i.e. the typical radius of interaction d. The rod consists, in other words, of a fairly large number p=l/d>>l of rigidly jointly monomer units. The position of each rod is given by the radius vector of its start, x, and a single orientation vector u. The produced structure has a steric distribution and orientation with a total density of the number of units n (x, u) and the probable density of coordinates and unit end direction ~ (x, u). The equations are formulated in the terms of the arbitrary probability operator ~ [14]:

~V/=S ff (x, u; x', u') ~ (x', u') d~x' dOu,/4~z,

(1)

in which dOu--the element of the corporal angle of vector orientation u. Analysis showed [9] distributions n and ~ to be linked in the equilibrial globular state b y the equalities

(2) in which the cross means an Hermitic coupling, while parameter A determines the standardization of V/. The entropy of conformation will equal S----Sn In [~'~/~] d3x dOu/4~-: ~ n In [~'+~+/~,+] d3x dOu/4~

(3)

A variety of chain types can be described b y various functions of g. We limit this here to two models in which the linking main chain rod is considered flexible and non-corporal. Model I is that of rigid branches [10]:

g =g0 (Ix-x'l)

(4)

a n d model I I that of rigid main chain segments [9]

g =go ([x-x'-lu'l)

(5)

The typical bond length between the rods is found from: go (x) d3x= 1;

~ x2 go (x) dax=a~

(6)

As both the chain models consist of identical units, the contributions of the reactions to the free energies at ao>>d for both the models having the same n(x, u) will be the same, i.e. will be described b y the single functional E{n}, so t h a t one can analyse the properties of the models by comparing them with each ~)ther.

114

A. V u . GROSBERG

After this it will be favourable to isolate the clearly angular part of the density distribution, namely to make n (x, u)----n (x) Wx (u);

S Wx (u) dOu/4.= 1

(7)

l~rom foimula (2) one easily finds for model I t h a t ~, (x, u ) = ~ + (x, u ) = ~ (x) w~ (u)

(8)

and for model I I ~+ (x, u)=~, (x+lu, - u ) = p +

(x) Wx (u)

(9)

B y inserting the results into the eqn. (3) for the entropy, one finds for model I that

in which ff is already the integration operator for the nucleus go in the normal three dimensional space. The result (10) has a simple meaning. As the direction in which the rods point in model I is not linked in fact by a linear memory, the entropy can be divided into the sum of two components, of which the first is for the entropy of packing in space of the bound starting points of the rods; it will naturally have the shape and meaning of the entropy for the simpler bead model [6-8]. The second component is the entropy for the independent, orientational distribution of the rods. An analogous formula for model I I is written as m--~-a

x--

1 • (x, u)dO.1

in which l~efl--the steric integration oyerator for the nucleus,

dOn,

gear (x, x')= S go ( l x - x ' - l u [ ) wx' (u') - an

(12)

Let us now analyse the result. We shall examine first the interior of a specifically large globule with a sterically uniform structure. The first components of formulae (10) and (11) will be zero in this area and the second will agree because of co(u)=co(--u) for steric regularity (uniformity). The free energies F ~ E - - T S therefore will also agree for both the models (see earlier text for E agreement). The sterie structures and all the thermodynamic properties of the globules will be the same in a bulk approximation where the macromolecules have rigid m a i n chain segments and rigid branches. Consideration must be given in the above analysis to the fact that the exam[ned phase diagrams of macromolecular solutions of semi-rigid chains [5"] were based on the possibility of neglecting the translational entropy of the chain units precisely as was done in the hulk approximation with the first componenta

!~Iacromolecule containing me~ogenie groups

115

of formulae (10) and (ll). All the other components of the free energy agree in our case with those described b y Khokhlov [5] so that the results of minimizing the free energy with respect to distribution w (u) can be directly taken from the latter article.

r~ rg8

r.

N

T~ Phase diagram: zone I--isotropic coil; zone II--stable, liquid crystalline globule, rectastable isotropic cell; zone III--stable hqmd crystalline globule, metastable isotropm globule; zone IV--liquid crystalline globule. One must not forget that the solution of semi-rigid maeromolecules will separate into the isotropic and liquid crystalline phases in a broad range of densities essentially above the/~-temperature [5]. (The 8-point is here that at which the second virial coefficient for the system of rods is inverted; this point coincides with the critical point of an isotropic solution consisting of infinitely long chain ibr example [4] within lip error limits [15]). This means that the respective equilibrial chemical potential as a function of density will have a range of negative values above the 8-point at larger densities, i.e. when still positive at low densities. I t therefore follows that globules with an orientational liquid crystalline order will form above the 0-point as found for the bead model [6-8], where the described shape of the chemical potential resulted in an insignificant dependence of the globular structure on temperature up to the globule-coil phase transition point. The latter has the features of a phase transition of first order and takea place, as said before, when Tn~O. Value 0 will depend on p when the nature of the forces acting upon the monomer units remains the same. An asymptotic formula O--8op/lnp [16] applies when p>>l. The Tn--p dependence will have the same functional shape in t h e case of a larger globule. The globular structure will be spherical in shape and the director of t h e local, nematic meso-phase will point in radial direction everywhere. We must remember that the coil-globule transition in short chains [9, 10] takes place below the 0-point. Quite clear is the fact that the transition temper-

116

A . ~ r u . GROSBERG

ature drops as the number of chain units N decreases, because the globule has a positive surface energy. No finite part of the chain can form thermodynamically favourable globule as the centre for a new globular phase directly at the temperature Tn; such formation requires the harmonized movements of all the units. The respective relaxation time will be obviously quite long ( r ~ N 2 for a Gaussian chain, as is known, while the exponent can differ from 2 where there is a reaction between chains). One must therefore examine the metastable isotropie state at T < T n . The smallest globules become more and more thermodynamically favourable when the temperature drops below Tn and will then be the seed for the globular phase; the relaxation time of the system in this t)hase will shorten correspondi ngly. This situation is typical of the kinetics of a first order phase transition. V.alue wx(u)~--I in the isotropic state and the nucleus geff (12) will depend only on the differential modulus of the arguments. The entropies (10) and (11) of both the examined models will therefore become those of the "bead" model. The respective typical length for model II will be 2

aeff~ S

y2

~3

2 , .2

geff (y) a y--~aott .

(13)

A phase transition of the chain will therefore be completed in the isotropic, metastable state as is typical for the bead model. It should be noted [7, 8] that this transition takes place slightly below the 0-point and makes possible a universal description within the terms of the second and third virial coefficients of chain units B and C. There are rods present as ~n_its in our case, so that [17] B,,,b~, in which b~12d, and ~----(v--0)/0, while C~ lada In 1/d. At p>>l we get

b/ao3 f<< 1 ;

(14)

and the transition to the isotropic globule in the case of model II [7, 8] will therefore be rapid below the 0-point and resemble closely a first-order phase transition. The transition picture for model I will depend on a0 and will be smooth (steady) in the 0-range for smaller values of a 0 ("dense comb"). The density will increase in the isotropic globule on reducing the temperature. The isotropie phase wiI1 clearly become absolutely unstable at the density which makes the isotropie state of the system with disjointed rods unstable, i.e. when the volume fraction ,ga~d21~a~ lip 2. From the results for the beads model [7, 8] it is easy to find the appropriate temperature, which must be below Chat for the existence of a metastable isotropic globule, and this will be feasible only when the number of chain units is sufficiently large

N~>p'~.

(15)

'The formula (15) got in previous work [7, 8] for the beads model does not con,aider the p-logarithmic factors as these are lying outside the accuracy limits of the theory.

Maeromoleeule containing mesogenic groups

117

The results are summarized in the phase diagram (Figure); they agree well with those of the numerous experiments carried out by others [18, 19]. There is actually no liquid crystalline phase created at small p-values (p>~l), but the typical parameters of (14) will be smaller and consequently the transition temperature from a coil to the globule will be lower relative to the 0-point [7, 8]. Any further p-increase will cause the phase transition temperature to increase more rapidly than the 0-temperature, and the transition at p>>l will be accompanied by an orientational ordering and will take place above the 0-point. Such an unsteady state is the main result got by El'yashevich and Skvortsov [18]. A more detailed comparison of the theory with the numerous experimental results must bear in mind that there was no length fixed for the rigid parts [18, 19] and that the determinations were made for an equilibrium. Please note in conclusion that any observation of the above phenomena must accept exceptionally rigid measures so as to avoid any chain aggregations and precipitations. The author thanks I. M. Lifshits and A. R. Khokhlov for valuable discussions. Translated by K. A. A~r_~r~

REFERENCES

I. S. ¢P. PAPKOV and V. G. KULICHIKHIN, Zhidkokristallieheskoye sostoyanie polimorov (The Liquid-Crystalline State of Po]ymers). Izd. " K h i m l y a " , 1977 2. V. P. SHIBAYEV and N. A. PLATE, Vysokomol. soyed. A I 9 : 923, 1977 (Translated m P o l y m e r Sei. U.S.S.R. 19: 5, 1065, 1977) 3. A. R. KHOKHLOV, Vysokomol. soyed. A21: 1981, 1979 (Translated in Polymer Sei. U.S.S.R. 21: 9, 2185, 1979) 4. A. R. KHOKHLOV, Vysokomol. soyed. A20: 2754, 1978 (Translated in Polymer Sei. U.S.S.R. 20: 12, 3084, 1978) 5. A. R. KHOKHLOV, Vysokomol. soyed. B21: 201, 1979 (Not translated in P o l y m e r Sei. U.S.S.R.) 6. I. M. LIFSHITS, Zhur. eksp. i teor. fiz. 55: 2408, 1968 7. I. M. LIFSHITS, A. Yu. GROSBERG and A. R. KHOKHLOV, Rev. Mod. Phys. 5 0 : 683, 1978 8. I. M. LIFSH1TS, A. Yu. GROSBERG and A. R. KHOKHLOV, Uspekhi fiz. n a u k 127: 353, 1979 9. A. Yu. GROSBERG, Vysokomol. soyed. A22: 90, 1980 (Translated m Polymer Sei. U.S.S.R. 22: 1, 1980) 10. A. Yu. GROSBERG, Vysokomol. soyed. A22: 96, 1980 (Translated in Polymer Sci. U.S.S.R. 22: 1, 1980) 11. P. G. DE GENNES, Phys. Letters A38: 339, 1972 12. P. G. DE GENNES, J. Phys. Letters 36: 55, 1975 13. I. M. LIFSHITS, A. Yu. GROSBERG and A. R. KHOKHLOV, Zhur. eksp. 1 teor. fiz. 71: 1634, 1976 14. P. FLORY, Statmtiqheskaya m e k h a n i k a tsepnykh molekul (The Statistical Mechanics of Molecular Chains). Izd. "Mir", 1972 15. A. R. KHOI(HLOV, J. Phys. 38: 845, 1977

118

L . B . SOKOLOVgt al.

16. T. M. BIRSHTEIN, A. A. SARIBAN and A. M. SKVORTSOV, Vysokomol. soyed. A17: 1962, 1975 (Translated m Polymer Scl. U.S.S.R. 17: 9, 2260, 1975) 17. L. ONSAGER, Ann. N. Y. Acad. Sei. 51: 204, 1971 18. A. M. YEL'YASHEVICH and A. M. SKVORTSOV, Molek. biol. 5: 204, 1971 19. T. M. BIRSHTEIN, A. M. YEL'YASHEVICH "and L. A. MORGENSHTERN, Blophys. Chem. 1: 242, 1974

:Polymer ScienceU.S.S.R.Vol. 22, pp. 118-123. 4{~ Pergamon Press Ltd. 1980. Printed in Poland

0032-3950/80]0101-0118507.50]0

SECONDARY REACTIONS AND THEIR EFFECT ON THE IRREVERSIBLE POLYCONDENSATION WITH A FEED* L. B. SOKOLOV, Yu. A. FEDOTOV and N. I. ZOTOVA All-Union Synthetm Rubbers Research I n st i t u t e

(Received 25 October 1978) The polycondensations of m-phenylenedlamine (m-DA) with the dichlorides o f t iso- and tere-phthalie acids were investigated while one of the monomers was slowly metered and secondary reactions were taking place. The features of the process mechanism were noted, and the rate constant of the polyamidation was determined under consideration of the effects of secondary factors due to the feeding method.

AN IMPORTANT feature of the irreversible polycondensation is the possibility of producing polymers of large molecular weight (mol.wt.) from a non-equimolar ratio of the original monomers by slowly adding one of them [1]; such a process was most successful when the reaction medium was inert to the reagents. Where secondary reactions are unavoidable in actual polycondensations, the respective mechanisms can vary and this makes them important. We are l~ere investigating the irreversible polycondensation process with a slow metering of the monomer taken in excess; the systems involved are subject to fairly intense secondary reactions. We selected as one of the emulsion systems that of THF-water-sodium carbonate. The presence of the alkaline, aqueous phase and some miscibility of the phases causes the progress of a hydrolysis of the acid dichloride as secondary reaction. The work involved a typical example of an irreversible process, namely the m-phenylenediamine (m-PDA) polycondensation with a mixture of iso- (80%) and tere-phthalic (20%) acid dichlorides (PADC). The main principles of the * Vysokomol. soyed. A22: No. i, 105-109, 1980.