Use of continuum model for calculating polymer melting characteristics

Polym~ Sclence U.S.S.R. Vol. 28, No. 12, pp. 2819-2827, 1986 Printed in Poland

0032--3950/86 $I0.00+.00. © 1987 Pergamon Journahl Ltd.

USE OF A CONTINUUM MODEL FOR CALCULATING POLYMER M EI,TING CHARACTERISTICS * A. L. RABINOVICHand V. G. DASHEVSKII(dec.) Institute of Biology,Karelsk Branch, U.S.S.R. Academyof Sciences A. N. NesmeyanovInstitute of Heteroorganic Compounds, U.S.S.R. Academyof Sciences (Received 5 June 1985)

The conformational entropy and the conformational free energy of melting of polymethylene and its oligomers are calculated on the assumption that the spectrum of the conformationsis continuous.The estimatesof the intermolecularand intramolecularcontributions to the entropy (24 and 76 % respectively)and the enthalpy (60.5 and 39.5y. respectively)of melting of polymethyleneare in agreement with experiment. The feasibility of theoretical calculation of the melting points of polymersis discussed. A STUDY of the polymer melting process within the framework of various approxima~ tions leads to a deeper understanding of the nature of the crystalline and amorphous states of polymers, and to a clarification of the extent of the influence of stoichiometrie structure and conditions of internal rotation of chain molecules on their macroscopic properties. The structure of the step-by-step theory of polymer melting encounters known difficulties connected with the necessity of taking into account all the factors affecting the process: chain flexibility, intermolecular interactions, structure of and defects in amorphous and crystalline phases, and cooperative nature of the factors [1-3]. The melting of polymers is assumed to involve two stages [4-8], i.e. expansion of the system with rupture of intermolecular bonds, but with preservation of the conformational states of the chains, and subsequent disordering of the chains (Bershtein [9]) observed both stages separately in studying superoriented linear PE fibres by DSC. Calculation of the conformational contribution to the changes in thermodynamic functions on melting in a rotational isomer model [1, 10, 11] is based on the idea of the presence of a strictly fixed, extended chain conformation in the crystal and of a mixture of rotational isomers in the melt; vibrat6ry motions in both phases and their anharmonicity are not considered. It is known [12, 13] that, for example, even n-alkanes of chain length N = 22-40 and odd n-alkanes of chain length N = 9-43 at temperatures 3-5 K less than the melting point pass into the rotational-crystalline state; the angles of internal rotation ~ of the chain in the crystal do not thus remain constant, but change noticeably [14--19]. In particular, investigation of C2s, C27, and C29 crystals by calorimetry and I K spectroscopy showed [16] that the proportion of nonplanar conformations increases * Vysokomol. soyed. A/g: No. 12, 2537-2544, 1986. 2819

2820

A. L. RABINOVICHand V. G. DASHEVSKII

rapidly close to the temperatures of the transitions between the solid " p h a s e s " ; in the highest t e m p e r a t u r e phase o f the C29 crystal a b o u t half the molecules are n o n p l a n a r . The c o n f o r m a t i o n a l c o n t r i b u t i o n to the e n t r o p y a single low t e m p e r a t u r e 0 - t r a n s i t i o n attains the value 0.42 J / m o l e . d e g [16]. I n this paper the m e l t i n g characteristics are calculated o n the a s s u m p t i o n that the spectra o f the c o n f o r m a t i o n s are c o n t i n u o u s ; the m a t e r i a l studied is polymethylene ( P M ) a n d its oligomers.

Model and method of calculation. It is assumed that the thermodynamic functions for the molten state of PM are equal to the values for the free chain under 0-conditions. As is known, such agreement for geometrical chain dimensions was established using small angle scattering of neutrons and X-rays for various polymers [20], including PM and n-alkanes [21-25]. The conformational energy Ub ( ~ , ~l+ 1) of the pentane fragment--(CH2)5-, entering additively into the total energy ~of the chain under 0-conditions was calculated by the method of Rabinovich and Dashevskii [26] Calculation of the statistical integral Z, based on the classical "flexible" model of a macromolecule [1, 27] is similar to calculation of the integral I in another paper by the latter authors [28] (with substitution, however, of the "mean exponents" [28] in the 60 × 60 matrixes by the statistical weights of each of the 3600 "states" of the map and elimination of the multiplier (,j~)N-3 in I [28]). It is ~tssumed that in the "crystalline" phase folded chain conformations are realized, in which torsional vibrations of the angles in the vicinity of the trans-position of each C - C bond are obtained. The allowed states of each pair of angles (~l, 0~+1) restricted the region of amplitudemaz close to the minimum (180, 180°) of the conformational map of the - ( C H 2 ) 5 - group; the quantity ~m,x effectively allows for the effect of intermolecular forces, and was considered to be a parameter which could be varied. The statistical sum of the chains in the crystalline phase was calculated formally by the above described method (since in the matrixes of the statistical weights only the elements corresponding to the vicinity of the total (combined) minimum are non-zero). The conformation of a molecule in a "crystal" thus corresponds to a worm-like chain; the degree of rigidity of this chain is evaluated below. The state corresponding to the maximum extension attainable in the given conformational model is selected from the standard state for calculating In Z in computing the conformational free energy F = - - R T I n Z and the entropy S = R In Z+RTdln Z/dT for both phases. The cal,culations were made for N=6-205 over the range 40-500 K at a spacing of ,JT=2, 10, or 20 K. The derivatives with respect to T were calculated from the Lagrange equation [29], numerically ,differentiated at five equally spaced points, while an expression was used for the derivatives at the centre points which gave the minimum error [29]. The linear plot of In ZN against N at T=constant was used [11]: the value of ZN was calculated accurately for N=6--21, following which, to speed up the procedure linear extrapolation of In ZM= C1 + C2 N was used. The constants C1 and C2 were calculated by the least squares method, using data for N = 12-21. Accurate control calculations of 1 In ZM (N=6-205) at several fixed temperatures I~ft the results almost unchanged. Conformational entropy of melting. Figure 1 shows the conformational entropy of the transition Intra lntra lntra AS~_k/N=[S~ -S~ (~ma~)]/Nper mole of CH2 units (linkages) as a function of the parameter ~ma~at the melting point (N--, ~ , Tm~,~411 K [30], the indexes "a" and "k" applying to the amorphous and crystalline phases respectively, and "intra" being the intramolecular contribution). Intra Estimates of the conformational entropy of melting ASmp /N obtained (with certain assumptions) experimentally and forming the "corridor" on Fig. 1 are reproduced at ~m,x~ 19 + 1° = ~0 ; the asymlntra Intra ptotic value of the conformational entropy of melting zlSmp /N=-S~ --SkIntra ~'o)]/N with increase in N is attained fairly quickly on calculation. This quantity has been calculated repeatedly within the framework of a rotational-isomer model [5, 8, 33-35] with various assumptions on the conformation of the chain in the melt (energies and number of conformers). Analysis of the results (Table in [33] and Table 8.7 in [36]) shows that the

Use of continuum model for calculating polymer melting characteristics •

Intra

2821

•

difference .4Stop / N Is determined mainly by variations of the parameter AEt., =. Eo [I I], asreement between theory and experiment was attained to some extent by happy selection of the statiBtical weights.

Calculations based on the assumption of a continuous spectrum of the conformations include both a rotational-isomer mechanism of disordering in the melt and also torsional vibrations of the torsional angles (with allowance for anharmonicity) in both phases. The intramolecular potential energy surface is given by selection of the atomatomic potential functions [26], and the only parameter of the crystalline potential which is not given directly is gm,x. However, refinement of the results is not expedient without detailed analysis of the structure of both phases, since a contribution to AS~-~"[N can be introduced, for example, by entangling of the chains in the melt [37], crystal defects, and other factors. The problem of the true form and methods of calculating the field of the intermolecular forces requires further study. Finally, the difficulties in showing the relation between "theoretical" and "experimental" data have not as yet been overcome [36]. The amplitude go, obtained (within the framework of the model) obtained under conditions where the experimental value of AS~pr=/N, is reproduced is in agreement with the results of a theoretical study of other consequences of intramoleeular motion in the crystal. Thus, Yamamoto [38] used the Monte-Carlo method in a continuum model to calculate the mean square dimensions of the chains of C6-C3o n-alkanes placed in I00

A

2OO N

I

B

1"00

50

lOON

I

10

I0

0"98

5

Ii L~, 1 II 1

,

¢p,.~ ,deg FiG. t

0'96

h/v/do FIG. 2

FIo. 1. Conformations entropy of transition A of PM (N--* o0) from the "crystalline" state into the amorphous state (1) at T=411 K as a function of the amplitude ~.~z of torsional vibrations in the chain in the "crystal"; the cross-hatched corridor is constructed from experimental assessments of the conformational entropy of melting of PM, J/deg.mole (CH2): 7.7 (T=410.7 K) [7, 39], 7.53 [6], 7.49 (T=413 K) [31], 7.79 (T=414.6 K) [32]. The conformational entropy of melting of PM B as a function of N at T = 411 K (2), A -= (S=~°tras/nt")/N, B - [S.~°t'=- S=~°t'=(¢o)]/N, J/deg" mole (CH2). FIG. 2. Ratio of mean distance hN= ((h2))t/2/(N - 1) on a repeating unit of "persistent" P M chains simulating PM fibres in the crystalline states (calculation at ¢m=I= ~0, T = 411 K), to the maximum projection do = l sin (0/2) of a linkage on the axis in the trans-position.

2822

A.L. RABINOVICHand V. G. Dmi-mvsKrt

the "crystalline" symmetry potential of a circular or elliptical cylinder, and gives an interpretation of the experimental data in accordance with longitudinal shortening of the period of the P M fibres on heating. It was found that the half-width of the peak of the distribution function P(tp) of the'angles of internal rotation around the tram-state in the C3o chain is, for different conditions, 12-23 ° for a circular potential ( T = 300 K; this increases by a factor of 1.2 at T = 550 K) and 8-12 ° for an elliptical potential ( T = 303-373 K) [38]. According to data from dynamic tests with a computer [39, 40] these deviations of the torsional angles in the crystals of n-alkanes amount to tens of degrees. In order to explain the degree of elongation of the conformations of the PM molecules in the "crystal" undergoing simulation by the Monte-Carlo method [41] an additional calculation is made of their geometrical dimensions. To do this, the conformational -(CH2)5- map, with the forbidden states outside the vicinity of a certain amplitude ~c(~,> ~o) close to its centre (180, 180°) was separated using the method given by Dashevskii and Rabinovich [41] into 1600 unequal cells of a 40 x 40 network; 156 cells located at the perimeter of the map were excluded since all their areas corresponded completely to the forbidden sections. By varying ~, on separation [41] the best correspondence was attained between the form and dimensions of the region occupied by the remaining "internal" cells of the network and the above used region ~o (although it is impossible to obtain complete agreement because of the differences in principle of the separation methods [28, 41]. According to the scheme in [41] the generated conformations are realized finally only in 1444 internal cells of the network obtained. Figure 2 shows the resultant relation between N and hN/do, which can be compared with the relative change d/do of the projection of the C - C bond in the direction of the axis of the molecule, where hs = (

)½/(N - 1), do = l sin (~9/2) ( is the mean distance between the ends o f the chain, l is the bond length, 0 is the valence angle of the chain, and T = 411 K). Extrapolation of the experimental data given by Dadobayev and Slutsker [42] with respect to the longitudinal shortening of the interplane distance in the PM crystal as a function of temperature, the curvature of which is related directly to T, gives a value of the order of about 0.99 for d/do at T=411 K. Accordingly, on a scale of several tens of units (Fig. 2) the simulated chains imitate real PM fibres in small crystals. Calculations on the persistent line " a " for the molecules gives a similar result: for example with a 72 unit chain evaluation of a at T=411 K gives a value a ~ 4 0 nm. Free energy. The free energy (per mole o f CH2 units) in the crystalline and amorphous states of a system can be represented by the sum of the intermolecular "inter" and conformational contributions Fk/N=F~,inter/N+Fdintra/N~ J~a/l¥-~a~ ~. ~-,inter//v . . . . . .1_/~ ~intra.~T/IV. The melting process, as a first order phase transition [1] takes place at the point of intersection of the curves Fk(T)/N and F.(T)/N; in adjacent regions each branch is unstable It is then found that intra

[(F,

-

F ~ n t r a ) / N a_ t ~inter

-~-.

g7 inter~ / AT-1

----k

:/"Jr=tin,=0

(1)

On calculating -~/~intr"= - R T I n Z o and "kr'intr"= - R T l n Zk(tPo), it is found from (1) that t~ f / f , ,inter - kF inter~/A/'I --Al~'interlAT J/,,Jr---r=p=,~,mp t,,. The curves F2ntr"(T)/N and Fl~ntra(T)/.N for

Use of continuum model for calculating polymer melting characteristics

2823

N = 205 are constructed from Fig. 3. Assuming that the asymptotic characteristics FIN for PM at N,-,200 are attainable in practice, then Fig. 3 at T=(Tmp)=xp gives the increment (Fiantra-F~ntra)/gV,and eqn. (1) gives the desired change in the intermolecular free energy. Using the experimental values of T~p for C~o-Cloo n-alkanes [36], the branches /M1 z¢ FIN were constructed similarly and the increments r/L-qntra t~r, - - "Lk' q n t r a ~)/~,Jr=r=p of these chains were found. The changes of free energy of intra- and intermolecular interaction on melting of CN ethylene oligomers are shown in Fig. 4. On changing eqn. (1) to the form Trap : r(14rintra LK'='=a

cint©r,,-I~zJnmp/zlSm p )J

1../int ra "t _L ( I . l l n t ©r ~ r i n t ©r,t-I / r - ( c i n t ra ~ i n t r a ' ~ ~_ ( ~ f n t e r ~Z~=k ) T ~'¢'¢a --'=~k /J/L\Ua Igk }T~Oa --gk

(2) and drawing on the experimental values of ,dHmp/N for PM (4.11 kJ.mole [36]), and comparing the intra- and intermolecular contributions to the enthalpy ziHmp/N and the entropy zlSmp/N of melting of PM (at N=205), the results obtained are: (H=l"tr= -H~"t'")/N=l.625 kJ/mole (calculated), and (N2"l'~-N~"t'~/N=2.485 kJ/mole [from eqn. (2)], or 39.5 and 60.5% of the enthalpy of melting respectively; (sinter-stnt©r)/N = [ill2 "t=~- H~"t=~)- (Fd"t=~- F~"t=~)]/NTmp= 2"398 J/K'mole, i.e. intermolecular interaction provides 24 % of the change of entropy of melting of PM, and intramolecular interaction 76% (Fig. 1), in accordance with experimental assessments [36]. 0

e00

~o

T,K

A

m

-2 . . . . . .

|

i

_

-200N

-4 F/N

kJ/molo

-2 B F~o. 3

FIo. 3. Conformational free energy of PM F~atr=(ffo)/N (1) and

Fio. 4

F~.*t"/N (2);

calculation for N=205.

FiG 4. Change in intra- (I) and intermolecular (II) free energy on melting of ethylene oligomers CN. The following values of T~p ,xp for n-alkanes (N in brackets) [36] were used in the calculations: 1 - 2 4 3 . 5 (10); 2 - 3 1 0 (21); 3-339.3 (31); 4 - 3 5 4 . 5 (40); 5 - 3 6 5 . 3 (50); 6 - 3 7 2 . 4 (60); 7-378.5 (70); 8 - 383.5 (82); 9 - 388.4 K (199). It is assumed that T 2-°s= 411 K. A = [(F.~ * l ' - F~°t")/N]r. r~=p, Int©r later N B = {[F~ - F~ (~o)]/N}r- r,,p k J/mole.

Calculation of melting point. Consecutive calculation of Trap for a polymer should evidently include calculation of the components of eqn (2) or the curves Fk (T) and F=(T). The difficulty of allowing for all the melting factors led to the development of semiempirical approaches, e.g. the method of extrapolation of the melting points of low

2824

A.L. RAerSOVlCH and V. G. DAsi-mvszn

molecular homologucs, the use of the correlation between the melting point and crystallization from the melt, or glass transition,correlation between energy of cohesion and density [l, 36]. In one of the firstquantitative theories of melting [l, 2] the main attcntio~ was given to the role of chain flexibility.The concept of the realizationof only unique trans. conformations for all the units of a chain in a crystal and the presence of mixtures of rotationalisomers in the melt was put forward. The intcrmolccular entropy of melting was considered to be negligibly small compared with the conformational entropy, and the entropy of cohesion for chain interaction served as the only factor for chain interaction, Tmp being reduced to zero when this was equated to zero. Flory also [4] conrtccts melting only with the flexibilityof the macromolecules, the intermolccular energy being assumed to be unchanged. Thus, a crystallinepolymer is melted when the proportion of trans-isomcrs in the chain becomes less than 37 ~o. However, as shown by Volkcnshtcin [3] numerical assessments of Trap, in spite of the results obtained by Flory [4], and in spite of the reasonableness of the assumption introduced and allowance for the cooperative nature of the melting process, differwidely from experiment. Allowance for torsional harmonic vibrations within the frameworks of a lattice type model [4] (this being in fact a rotational isomer model) and on the assumptions given by Volkcnshtcin [2] on the conformations of chains in a crystal and in a melt, led Volkcnshtcin [I] and Volkcnshtein and Ptitsyn [2] to a reasonable value of Trap for PM at realistic values of the parameters. It was shown also [3] that Trapexp could be reproduced, and without allowing for rotational isomerism, but for fairly small values of the vibration frequencies in the well, corresponding to trans-conformations. In the papers [3, and 4] by Volkenshtein and Flory respectively the change in intermolecular free energy was not taken into account. It is submitted that papers [2-4] demonstrate the importance in principle of including each of the components of (2) in the calculations, since in the absence of any component Tmp exp must be obtained by varying the remaining terms within fairly reasonable limits. In this plan the work of Sundararajan [35] is significant, in which an attempt is made to use for calculating Trap the fact that in a first order phase transition the first derivatives of the free energy with respect to temperature at the melting point should undergo discontinuity (when N---,c~), and the second derivatives have a maximum. In accordance with this assumption,

dS/dT=R(2dlnZ/dT + Td21nZ/dT 2)

(3)

or, after division by Td 2 In Z/dT 2, the function

f=[2(dlnZ/dT)/(Td21nZ/dT2)'l + 1

(4)

which is interpreted by Sundararajan [35] as the melting point of the polymer. However, instead of a complete statistical sum of the system, in eqns. (3) and (4) an expression for Z of the amorphous phase only (more accurately the chains under 0-conditions [35]) is used. Moreover, the branches Fa(T) or Fk(T) in themselves have no special features at the point T~-Trap' and the physical significance of the maxima on curves

Use of continuum model for calculating polymer melting characteristics

2825

0"3

0"2

0.1

0

I

I

I

100

200

300

m''-:i

I

.. [

#00

T,K

FXG. 5, Function dS~,°'"/dT for polymethylene chains for values of N of •-20; 2 - 4 0 - 3 - 6 0 , 4 - 80; 5-100; 6-120; 7-140; 8-160; 9 - 1 8 0 , and 10- 205 as a function of temperature; continuous conformational spectra.

f, rel.un. O.q O.Z j zoo

0 -0.2 -O.q FxG. 6, Function f [from eqn. (4)] as a function of temperature for PM; N=,205, continuous conformational spectra.

2826

A . L . RAnI~oVlCHand V. G. DASHEVBKII

(3) and (4), generally speaking, is not connected with T,np. By varying the matrix elements of the statistical weights Sundararajan [35] nevertheless obtained agreement between the position of the maximum on curve (4) and Trapexp for 11 of the most widespread polymers; in five cases, including PM, the weights recommended by Flory [11] were used in calculating the geometrical characteristics of the unperturbed chains. To investigate the problem the functions (3) and (4) were calculated in the continuum model (second derivatives of log Z0 with respect to T calculated by the same method as the first). The results are shown in Figs. 5 and 6: the desired maxima of both functions were displaced relative to Trapexp by almost 300 K. Thus, the point of inflexion of the function S,j*tra can be combined with Trap~x~ evidently only on special selection of the parameters of the rotational-isomer model; at "real" statistical weights the model corresponds only for specific polymers and is determined by the small contribution of the nonconformational components or by the compensating action of various factors. It is interesting that the case of PM (and polystyrene) caused Sundararajan [35] certain difficulties, since the function f was negative over the whole temperature range (35] and was replaced by the modulus f. It is reasonable to suggest that in the work cited the real maximum of f for PM, if it existed, was displaced from Tmp~p' so that it was outside the reasonable temperature region in which the search was carried out; the maximum of however at T~420 K [35] could be a consequence of fluctuations resulting from errors in the numerical differentiation of In Z (fluctuations are also present over the whole temperature range, but these are of magnitude less than that of the main peak of f by a factor of 102-104). It is noted that a formal mathematical maximum of dS/dTalready occurs in a system of particles having two (or three) different energy levels (and polymers widely used in physics [7, 43]). The authors with to thank T. M. Birshtein for comprehensive discussions on the problem as a whole and also individual aspects of the work as it proceeded.

Ill

Translated by N. STANDEN'

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Use of continuum model for calculating polymer melting characteristics

2827

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