General mechanism of the β transition in polymers

PolFmer ScienceU.S.S.R. Vol. 27, No. 11, pp. 2743-2757, 1985 Printed in Poland

0032-3950]85 $10.00+ .Off © Pergamon Journals Ltd.

DISCUSSION GENERAL MECHANISM OF THE fl TRANSITION IN POLYMERS* V. A . BERSHTEIN a n d V. M . YEGOROv Ioffe Physico-chemical Institute, U.S.S.R. Academy of Sciences

(Received 19 May 1984) The results of DSC investigations show that in polymer relaxational processes the fl transition is a general phenomenon common to noncrystanizing solids. A relationship has been found between thermoactivation parameters of the transition and molecular characteristics i.e. the average length of a molecule, the length of the statistical Kuhn segment, the cohesive energy and the potential barrier of internal rotation. It is concluded that the act of ,8 transition is comparable to acts of relaxation in liquids, and takes place in sites of less dense packing. For linear nonrigid polymers the act consists in the rotation of a chain fragment similar in length to the Kuhn segment involving the surmounting of preferentially intermolecular barriers with the participation of T-G transition; the kinetic unit corresponding to the fl transition is related to the segment of intermolecular mobility in polymers.

THE TERMSB transitiont [1, 2] or T< Ts relaxation [3] refer to a relaxational transition that is cosest to the glass transition temperature TS of a polymer and that satisfies the Arrhenius equation for " j u m p " frequency v ml013~2 exp (-Q/RT). It is found even in simple vitrified liquids [4], and is thought to be c o m m o n to any solid that has a disordered structure [5]. Authors differ in their opinions regarding the source of the # transition in polymers: it has been associated with the movement of 1-2 monomer units [2, 6] with the movement of short chain fragments [1, 7] or side groups [7, 8], with movements within the latter [9], and sometimes with the presence of impurities such as water, monomer and some others [1, 7]. The problem of the nature of potential barriers to fl relaxation is also discussed by authors [5-7]. However, in recent years data have been obtain which show that the onset of rotational-translation displacement of main chain fragments plus side groups occurs precisely in the region of the fl transition (temperature Ta) and apparently appears in free volume sites. For instance, the results of low-frequency combinatorial light scattering experiments [10] as well as the results of low angle X-ray and light scattering in polymers [11, 12] along with data obtained by dynamic analysis of H bonds in deformed polymers [13, 14] appear to indicate that approximately at T a we have the onset of a major free volufne increase owing to (thermal) density fluctuations, as well as the onset of motion of chain fragments, whose length is equal to several units, together with rotation of the latter through angles wide enough to permit conformational transitions. Starting at Ta we have what appears to be the "suppression" of chain mobility in oriented polymers under mechanical loads [15]. Molecular regrouping ("physical ageing")in polymers accompanied by a change in their properties likewise occurs in the temperature interval Ta-T~ [16], starting in the region of the fl transi* Vysokomol. soyed. A27: No. 11, 2440-2450, 1985. 1" Regarding use of the term "fl transition" in the case of highly crystalline polymers, see below. 2743

18 16 20 30 22 20 18 38 58 50

Polydiphenylhydroxyphenylene (PDPHP) Polybutylmethacrylate (PBMA)

PS

PAN Poly-=-methylstyrene (PMS) Polycyclohexylmethacrylate (PCHMA)

Polyimide PI-DP-DADFE (PI-2) Polymetaphenyleneisophthalamide (PMPIPA)

4§ 7"5

12 9 8 7 3,5~

3 6-7

2"3 9 6 11 7 12 7 6

3-4 4-5 5 2'5 ~2 18 2

[80] [82]

[325 [33] [33] [32, 33] [801

[321

[33] [33]

[83] [32] [33] [33] [321 [32] [33] [32, 33]

[335 [32, 33] [33] [33] [335 [33] [33]

[321

Ref.

--j

]

130 85

42 50 130

32

36

90 40

50 14 25 15 30 18 30 32

25 17 20 34 70 6 60-70

8

E~, kJ/mole

473 (590 Hz) 430

340-370 290-330 448 (590 Hz)

330-340

300-330

350 280-290

300-360 (,8') 220 170-200 150 220-260 200-240 230-250 ~ 300 280-290

120-150 135-140 130 160 210-220 170-180 150-200 (,8)

140-170

Tp,K (at ~ 1 Hz)

DSC

[81]

DSC DSC [815 DSC

110+ 10 90-100 120 140+ 15 140 200 + 20

50-55 46 42+8 60+5t 55-60 60-65 75* 80_+ 12 90+8 95-100 92+8 100 96 110+ 10 110+10

100:2:10

[1, 84] DSC [19] DSC DSC [88] [1, 90] [11 [41-43] DSC DSC [1,361 [1,191 [1,9] [89] [1,86] [1, 85] [1] [15 DSC [88] DSC [1] DSC [26] DSC DSC

Ref.*

36+8 36+8 34 38+4 34+4 42-50 50-55 42t 42_+ 12 45-52

Q # , kJ/mole

.8 ( T < Tt)-relaxation

120

95-11o [78]

lO5

70 [79]

60

113 [91] 4o....45 38

57 [91]

38 15-20

30

Q~, kJ/mole

* D S C - data obtained in the present investigauon. ? Q~ values were mostly from monograph [46], but in the case of four polymers, from the papers cited. In the three cases indicated the Qp values were calculated by ourselveS on the basis of T~ values in the Arrh©nius equation from the " j u m p " frequency. § In view of the poor solubility of polyimides as well as the lack of data on 0-solutions we took theoretical evaluations of the length of Kulm segments borrowed from [80].

Polyimide PI-DPO (PI-I)

18 22 16 28 17 30 17 15

PA-6 PP Polydimethylacrylate (LAMA) PVF PVAc PVC PVAIc PMMA

15

46 14

14+2 11-14 14 14

20

S units

Lcn ths of Kuhn segments A x 10a, c m

SKI PDMS Polydiethylsiloxane (PDES) Polyhydroxyphenylene (P HP) PETP PTFE PC

PE

Polymers

VLSCOUS FLOW OBSERVED IN POLYMERS

CHARACTERISTICS OF THERMODYNAMIC STIFFNESS OF MACROMOLECULES, INTERMOLECULAR INTERACTIONS, f l ( T < ~ g ) TYPE RELAXATION AND NEWTONIAN

0 0 0 <

.<

g~ f3.

m

~e

.< .>

General mechanism of fl transition in polymers

2745

tion [17, 18]. It should be noted that when linear plots of log v(1/TB) in the frequency range 10 - 2 106 Hz are extrapolated to the area of the liquid state (frequencies of 10s-101° Hz) the plots describe relaxation processes in liquids satisfactorily [5, 19]. In view of this we surmise that molecular movements in liquids are similar to these in the solid-state fl transition. On the basis of experimental data published in papers [20, 21] we are now able to propose for linear polymers a general explanation of fl(T< Ts) relaxation whereby it is determined by the conformational mobility of chain sections that are commensurate with K u h n segments. The present paper sets out the experimental data substantiating our viewpoint along with a brief" discussion of the significance of this interpretation of the fl transition, in particular, as regards prediction of polymer properties. DSC experiments were run on a group of linear polymers, on vitrified low molecular weight liquids (see Table) and on four groups of oligomers, 1) ~t-methylst~rene oligomers with average degrees of polymerization n = 1, 4, 7, 14, 2 x 102 and 2 x 103 (the oligomers were synthesized via anionic polymerization [22] with M W evaluations based on exclusion liquid chromatography; well vitrified isopropylbenzene was taken as a m o n o m e r model); 2) styrene oligomers with n,~5, 9, 20, 2 × 10 a and 3 x 104 (using narrow f r a c t i o n s - PS standards in chromatography, Mw/Mn ~< 1. I); 3) PC oligomers with n ~ 1, 3, 4, 7, 130 and 460 (vitrified l , l - d i p h e n y l p r o p a n e was taken as a monomer model); 4) dimethylsiloxane oligomers, viz hexamethyldisiloxane (dimer) and products of PMS-6, PMS-10, PMS-30, PMS-100 and PMS-400 with n,~8, 16, 23, 50 and 100 respectively. DSC was used in this investigation to detect fl transitions, to determine T~' s and the average effective activation energies Q~. The test conditions were in keeping with frequencies v ~ 1 0 - 2 - 1 0 Hz. T h e DSC-2 Perkin-Elmer calorimeter was calibrated from melting points of acetone (178-0), ice (273'1), indium (429'7), lead (600.7 K) and in accordance with the heat capacity of sapphire. The procedure followed in our investigation of the fl transition was as follows. Samples weighing 10-30 mg were heated to T ~ Tg+ 100 K,.and were then cooled in accordance with one or other of these two regimes: at the rate of 1"25 d e g - m i n - 1 to T ~ T g - 1 0 0 K (annealed samples)or by quenching in liquid nitrogen (to 77 K). In the case of PC cooling was continued down to T ~ T g 300 K. Upon heating the samples in the calorimeter it was seen that the DSC curves for both annealed an d quenched samples have a step in the heat capacity zl Cp at Tg; the curves differed as to the character of the temperature range: from the beginning of the region of the fl transition (temperature T~) to temperature 7-2 ~ T + 10 K (Fig. la), i.e. in the area of structural relaxation ("physical ageing"). It is seen from Fig. l a that the region of the structural transition is also identifiable by preliminary deformation of the polymer. The area between the DSC curves corresponds to the enthalpy difference between the samples [17]. Retention or exposure ("ageing") of the quenched sample for 0.3-1 hr at T1 leads to an endothermic peak appearing on the DSC curve and having a TB m a x i m u m located close to thermal fl peaks when measuring is performed by standard relaxation methods. As a standard we took hardened ethylene glycol dimethacrylate ( D M E G ) , or, for more clear-cut identification of the fl peak, a quenched sample of the same polymer that h a d not undergone ageing (as an example of the differential curve, see Fig. lb). Retention at T~ corresponds to the initial stage of enthalpy relaxation taking place as a result of fl processes, while the endothermic peak on the DSC curve relates to the emhalpy increase owing to "defrosting" of these same processes. As the temperature and the time of ~geing increase, endothermic peaks on the DSC curve are obtainable at any temperature T ~ T ~ Tg [17, 23, 24]; the peaks characterize defrosting of molecular motion predominating at a given temperature [24]. Values of Q~ (in kJ per mole of units of movement were determined from the displacement of T~ (Fig. I b) while heating (at the rate v--0.3-40 deg. min a series of samples prepared in this way. The linear relations for v(l/T~) shown in Fig. 2 were the basis of calculation of Q~, using the formula

Rdln v

[25]: Qt~. . . . .

d(1/T~)

In this case a separate calculation was carried out to take account of the

effect of a methodological factor, viz. the temperature lag due to thermal resistance. It should be

V. A. BERSHTEIN and V. M. YEGOROV

2746

n o t e d that fl relaxation is characterized by activation energy dispersion, though according to Bershtein [26] the latter does not a m o u n t to more t h a n + 10-15 ~ on the most probable value of QB, which is within the limits of experimental accuracy in the present instance. A wide range of published information relating to mechanical and dielectrical relaxation processe~ were analyzed in the course of verification of relations between Qa and molecular parameters of substances, and the most reliable values of Qa were selected (see Table). It can be seen that the Q~ values we obtained on the basis of DSC are in satisfactory agreement with those obtained using standard relaxation methods.

~

a

b

t

•

2 3

I

{

I

I

31t7

330

350

370

T,/~

FxG. 1. DSC curves for polystyrene in Various states where v = 5 deg m i n - t (a) and differential curves of the fl peak for samples after ageing (b). a: 1 - annealed samples; 2 - q u e n c h e d samples; Y - q u e n c h e d samples after subsequent retention for 15 min at temperature T1; 3 - a f t e r prior deformation by compression to a 40 ~ degree of residual deformation; b: heating rates of 5 (1), 10 (2) and 20 deg min -1 (3). In addition, we included in our investigation the parameter of intermolecular interaction, the cohesive energy E~ (the values for polymers being calculated per mole of monomer units). Values of Ec were either borrowed from [27-30] or were calculated by the method described in [31], and lengths of the statistical K u h n segment A containing S m o n o m e r units were calculated. Values of S characterize the thermodynamic stiffness of the macromolecules a n d are evaluated in dilute 0solvents or at 0-temperatures [32]. The K u h n segmental lengths were taken from papers of other authors (see Table); to do so we also used values of ( ( ~ ) 2 / M ) ~ cited for m a n y polymers in [33]; here ( 7 ) * is the mean-square distance between chain ends in the unperturbed state, and M is the molecular weight o f a chain of average length. Where N is the n u m b e r of segments in a macromolecule o f contour length L, and Mo and 2 are the M W and the length of a single unit, we have 7 2 = N A z

General mechanism of fl transition in polymers M

~2Mo

Mo given in the Table.

MA

= L A = - - 2 A , i . e . theKuhnsegmentallengthA=

2747

• All the values of TB, Q~, Ec, A and S a r e

Determination of temperatures and activation ener#ies of fl(T < T~) relaxation. It is known that relaxation methods do not invariably allow clear-cut identification of the p transition: in the case of good molecular packing it may be largely "suppressed", and may sometimes be found to merge with the low temperature wing of the ct peak, more especially at high frequencies while for individual, polymers, such as P D M S and PDES it has altogether failed to appear. In u Fde,.q.rnm-~

!

3

25

3.7

3.9

72

7-5

q

7.s 70a/~ K"

FIG, 2. Relations between temperature of the B transition and the heating rate for PMS (1), hepta-c~methylstyrene (2), isopropylbenzene (3) and PI)MS (4). In our DSC study of fl relaxation as an Arrhenius transition closest to Tg it was found to occur in all the study objects (polymers, oligomers, simple molecular glasses) at temperature Tp that was ~ 20-100 K below T8 (in the case of PC, 220 K below Tg). For low molecular weight glasses the # peak is particularly close to Tg. Figure 3 shows examples of DSC curves with a fl peak and with a step in the heat capacity ACp near the glass-transition temperature. This result is in keeping with the idea expressed in papers [4, 5] that the fl transition, is a "precursor" of a cooperative process o f relaxation close to T~ (the c(transition) is a general phenomenon that is common to solids having a disordered structure.

~T

100

150

250

300

350

TjK

FIG. 3. DSC curves for isopropylbenzene (1), P D M S (2), polystyrene (3) and P C H M A (4). Heating rate 10 d e g . m i n - x.

2748

V. A. BERSHTEINa n d V. M. YEGOROV

Below we will be commenting on data relating to the fl transition occurring in individual polymers with a view to describing its characteristics more precisely, and to show that there is no reason why the p transition should be classed with processes that are unrelated to main chain movements. The intense relaxation peak at ~ 200 K (1 Hz) in P C H M A is assigned to fl transition due to movements within side groups (an " a r m c h a i r - a r m c h a i r " type conformational transition in the cyclohexyl ring) [7, 9]; the assignment of this peak has been verified experimentally. However a transition that occurs at ~ 3 0 0 - 3 5 0 K [34, 35] and was detected by DSC (Fig. 3) must be classed as the one that is an Arrhenius process nearest to T~. The tabulated values of TB and Qa for P C M H A refer precisely to this transition. On the other hand, the transition at ~ 2 0 0 K has been correctly assigned by Boyer to a specific 7 relaxation [35]. In PA-6 the ,8 transition at Tp~220 K is usually associated with the presence of water in the polymer in view of the absence of any clearcut fl peak when its internal friction spectrum is recorded immediately after annealing [1]. However, absorbed water is not primarily responsible for the fl transition. It is reported in [36] that the fl peak does not disappear during heating of PA-6, but undergoes a transformation (it widens out and is displaced towards low temperatures, towards the peak). This occurs as a result of disruption of the intermolecular H bond system. Subsequent storage of PA-6 in the absence of moisture leads to a recovery of the H bond system, and the original fl peak at ~ 220 K reappears in the dehydrated PA-6. In the case of PE and other polymers with a degree of crystallinity exceeding 40-50 ~ transitions denoted as ct, fl and ~ transitions are usually assigned to transitions that do not correspond to analogous ones occurring in amorphous and slightly crystalline polymers, which does lead to a great deal of confusion. A large amount of data have been published in this connection, b u t there still appears to be no agreement a m o n g authors regarding what temperatures should be assigned to T~ a n d to T < T~-relaxation in PE (see references cited in [37]). Fresh data obtained by a combination of several methods, including D S C [37, 38] very clearly show that relaxation at ~ 240 K should be assigned to Tg in PE (this is usually referred to by authors [1] as the fl transition), while the true fl transition ( T < T~-relaxation)relates, in fact, to the only clear-cut Arrhenius transition in PE that occurs at ~ 140-170 K, and is referred to as 7 relaxation in literature. Similar confusion arises in regard to relaxation transitions in other highly crystalline polymers (PP, P T F E and others). Values of TB and QB for the true fl ( T < Tg)-relaxation have been tabulated for PE and other polymers (see Table). Complex types of relaxation behaviour occur in P A N and PC. A relaxation region appearing in the interval ,-~350-400 K for P A N could not be resolved into fl and 7 transitions in [39, 40]. With the aid of DSC this was achieved (see [24]), and the QB and Ta values have been tabulated (see Table). PC having two "cardic" O atoms in the m o n o m e r unit possesses low-temperature plasticity, an unusual property for plastics. A region of intense fl relaxation is observed in PC at 150-200 K [41-43], which is far removed from Tg~420 K. In addition, there is an intermediate region of relaxation at 300-370 K [41, 43] which is particularly well defined in the case of quenched or deformed samples. The activation energies were determined by DSC for both transitions (see Table) and their Arrhenius character is thus revealed. Without doubt the onset of conformational mobility appears in PC at 160-200 K [44], and the other transition occurring at 300-370 K is closer to T~, so as an exception both transitions, designated as fl- and if-relaxation, are analyzed below. For siloxane polymers ( P D M S and PDES) it appears that fl transitions were first detected by DSC, and the Qa values were determined (see Fig. 3, and Table). In the case of P M P I P A ( T ~ 540 K) the relaxation peak at 200 K is assigned [45] to the fl transition, though relaxation also appears at ~ 4 5 0 K. "[his was detected and characterized by us by DSC, and was assigned to the fl transition (see Table). In P M M A the fl transition is attributed to inhibited rotation of side groups [l] or to movement of the latter with some participation on the part of the main chain [7]. However the results of direct experiments [10] point in this case to movement of chain fragments whose length is equal to several m o n o m e r units. The problem of the fl transition in polymethacrylates containing bulky alkyl radicals

General mechanism of fl transition in polymers

2749

and in other comb-like polymers in which internal platicization occurs calls for separate analysis; this will be the subject of a future communication. We would say only that the apparently anomalous behaviour of the ,8 transition in these polymers (the slight variations in the QB barrier when the side alkyl radical is lengthened) is a reflection of the fact (as revealed by the longwave IR spectra) that the movement of main chain fragments is unrelated to side radicals linked to the chain via "car dic" oxygen atoms [92].

a

h

100

~

~ 1

~ 30

~ 10 z

~., 1L7~

~1o~

720

~ 40

i 30

~

I

lO

)0 z n-

FIG. 4. Activation energies of the fl transition vs. average n u m b e r of m o n o m e r units: a - P S (light points) and PMS (dark points); b - P D M S ( 2 - Q , [51]); c - P C ( l - Q a , 2-Qa').

Relations between thermoactivation parameters of 1~ relaxation and molecular characteristics. Eyring obtained relations between activation energies of flow Q, for hydrocarbons and the length of their molecules; in this way he determined the segmental nature of flow of polymers, and estim a t e d the length of the rheological segment of movement in PE [46]. This method was likewise employed in our study of the 1~ transition. In Fig. 4 we have DSC plots of relations between the activation energy Qa and the average n u m b e r of m o n o m e r units in the molecule n for the above-mentioned oligomer-polymer series, viz for PS, PMS, P D M S and PC; in PC the relations of Qa(n) and Qa,(n) were measured. For PS and PMS the Qa values fit (within the limits of experimental accuracy) a single curve of Qa(n). The comparable course of corresponding relations can be found in [24] for transition temperatures Ta(n) and T~(n). The Qa(n) curves plotted by us are similar in shape to the Eyring plot of Q,(n) for a segmental process of viscous flow for polymers [46]: the energy rises until a definite chain length is attained, after which it becomes practically independent of n (Fig. 4). It can be seen that for PS and PMS the value of Qa is many times increased on going from m o n o m e r to an oligomer with n ~ 10. In the case of P D M S the value of Qa is significantly lower for the dimer only and is equal to 32-38 kJ/mole; for the other members of this series (8-, 16-, 23-, 50- and 100-mers) an inftexion on the Qa (n) plot appears in this case, i.e. for n ~ 5 ± 2 units. In the oligocarbonate series the energy Q attains saturation already at n ~ 2 - 3 , while Qa' does so at n = 4 (Fig. 4). For PS, PMS, P D M S and PC the statistical K u h n segments have S ~ 9, 8, 4-5 and 2 units respectively (see Table), so our data show that increase in the QB barriers as the chain is lengthened continues only up to the point where a critical length is attained, which is commensurate with the length of the K u h n segment. This rule for activation energies ought to hold equally, as a first approximation, tk;r average relaxation times in fl transitions: linear extrapolation of experimental plots of log v(l/Ta) or plots of log v ( I / T a) found in literature, to I / T = 0 gives a practically constant value of the pre-exponent Vo~ 10 ~a ~z Hz, thus substantiating ideas of the low enthropy Arrhenius character of the fl transition. In view of this finding in particular we developed a hypothesis [20, 21 ] of the source of fl relaxation in polymers: whereas in glasses, consisting of simple molecules, the act of fl transition may be defined in accordance with a presumed [5] rotational-translational displacement of a molecule, as the natural kinetic unit, into a " h o l e " we would say that in polymers (n > S) it involves an analogous movement of a " q u a s i - m o l e c u l e " - i . e . a chain fragment whose length approximates to the K u h n segment. The kinetic unit in a f l transition c a n n o t be significantly longer than the K u h n segment (where n ~ S Qa m const) nor can it be much shorter, e.g. equal to a single unit, as is sometimes

V. A. BERSHTEINand V. M. YEGOROV

2750

supposed [6, 8, 47].* This is not only at variance with the data in papers [10-14] but, in the latter case, low values of Qa for short oligomers would be attributable to low potential barriers to movement of terminal units compared with middle ones, though (for instance) Qa ~ const for the PC trimer and polymer (Fig. 4). There are also other results (see below) supporting a hypothesis of the segmental nature of the fl transition and opposing an idea that the kinetic unit in this transition could be a monounit. Figure 5a shows the experimental Qa(Eo) plot for a group of vitrified liquids and oligomers with n < S, differing markedly as to molecular structure and in regard to intermolecular interactions. This group includes, for instance, o-diphenylbenzene and decalin with rigid molecules in which no internal rotation takes place; also 5-methyl-3-heptanol with flexible molecules, and glycerol with a developed H bond system, and other polymers. It is seen that for the entire series of substances the elementary points fit a plot of Qa~(0.4_+0.1)Ec satisfactorily; here Ec for the oligomers refers equally to the molecule as a whole.

QI3, k,.7/mole

I 80

a .

t~O .k'7/m°/e • -I

200

~J" I 1 40

19~218

lO

6 7~ .''f

I

120 EK, k J ~ eo . . . ~ L,%,~2!

25"t

90

T / ~ "

.e,

17e 16 18oo,~ "22 . I_5~

IOO

i

"13 t

I 200

I

I

I

30

(2?,kEfm~,~

qOO E~'S,kJ/mole FIG. 5

I

90

FIG. 6

FIG. 5. Activation energies of the fl transition in simple molecular glasses, oligomers (a) and linear polymers (b) vs. cohesive energy or cohesive energy of Kuhn segment respectively, a: 1 - 2-pentanol [4], 2-isopropylbenzene (DSC), 3-5-methyl-3-heptanol [87], 4 - d e c a l i n [4], 5-1,1-diphenylpropane (DSC), 6-diethylphthalate [87], 7 - g l y c e r i n (DSC), 8 - o - t e r p h e n y l (diphenylbenzene) [4], 9-hexamethyldisiloxane (dimer), 1 0 - tetra-~t-methylstyrene, 1 1 - pentastyrene (DSC); b: 1 - PE, 2-SKI, 3-PDMS, 4-PDES, 5-POP, 6-PETP, 7-PTFE, 8-PC, 9-PA-6, IO-PP, 11-PMA, 12-PVF, 13-PVAc, 14-PVC, 15-PVAIc, 16-PMMA, 17-PDPOP, 18-PBMA, 19-PS, 2 0 - PAN, 21 - PMS, 22 - P C H M A , 23 - PI-1, 24 - PI-2, 25 - PMPIPA. FzG. 6. Relations between activation energies of the fl transition in polymers and Newtonian viscous flow of their melts (numbering is the same as in subscript to Fig. 5). * The authors of [47] relate fl relaxation in styrene-butyl methacrylate copolymer melts (a joint =,//-process) to movement of the monounit, seeing that QB is not a function of the copolymer composition. However, this may be accounted for in terms of practical equality of Kuhn segment cohesive energies (Ec S ~ 2 9 0 kJ/mole) for PS and P B M A (see Table) which, as we will show below, accounts for equality in the Qt~ values of these two polymers.

General mechaniem of fl transition in polymers

275l

It is remarkable that this equality is akin to the classic Eyring formula [46] for the activation energy of flow for simple liquids Q, ~ (0.3 + 0.05) AHe, where the heat of evaporation AHe--E~ + R T Ec. The Eyring formula shows that the potential barrier to movement of a molecule in a medium o f other similar molecules amounts to a fraction (approximately one-third) of the total energy of' intermolecular interaction (E~). The fact that the formula is similar for Qa therefore supports and substantiates the idea that the fl transition in glass, which consists of small molecules or short oligomers, is a "liquid-like" act of displacement of the molecule as a whole, and points to the intermolecular nature of the fl relaxation barrier in this case. The found correlation is also in good agree-, ment with the linearity of log v(I/Ta) plots over a very wide frequency range, including the vitri-. fled state and relaxation in melts [5, 19]. It is also of fundamental importance that potential barriers to Qa and Q, proved to be similar for polymers as well: it is seen from the Table and from Fig. 6 that the difference between the values is usually no more than 10-20~o, which could be ascribed to experimental error and to some dimi.n u t i o n o f intermolecular interactions with temperature, particularly above the flow temperature '[48, 49]. The latter effect is b o u n d to be especially marked in polysiloxanes: this is a p p a r e n t frorn the abnormally high coefficient of thermal expansion in P D M S above Tg [50]. It is this, apparently, which led to the observed shape of the Q,~(n) plot which (even in this anomalous case) is practically ihe same as for Q~(n) (Fig. 4b). The good agreement between QB and Qn for the polymers is yet a n o t h e r pointer in favour of the liquid-like character and segmental nature of fl relaxation in polymers, as well as supporting the idea of similarity between elementary acts of fl transition and of t h e viscous flow in a polymer melt. Next, it was found that in the case of polymers there is no unambiguous relationship between QB and energy E¢ referred to 1 mole of monomer units. Whereas it is conventional to regard a linear polymer as one that consists of quasi-molecules ( K u h n segments) as kinetic units, it would appear from the analysis carried out for 25 polymers (see Table) that a relationship of the type illustrated in Fig. 5b is valid, viz Q a ~ ( 0 ' 3 +0'05) EcS+B, where Qa is expressed in kJ per mole of segments, a n d the term B ~ 15 kJ/mole makes but a relatively small contribution to the potential barrier for the fl transition. The equation for energy QB in polymers, referred to the quasi-molecule, is akin to expressions for QB and Q, in simple molecular glasses and liquids: the main contribution to potential barriers to r/ relaxation in polymers is likewise made by intermolecular interactions. The sole difference here lies in the presence of the term B, the value of which is equal to the internal rotation barrier in flexible polymers. The latter is understandable if it is the case that the act of fl relaxation comprises a conformational (T-G) transition. This is substantiated, in partciular, by spectroscopic data relating to change in the intensity of conformation-sensitive absorption bands for PS in the 500-600 c m - t region during fl transitions [14]. It is therefore natural to surmise that of the large n u m b e r of main chain torsional-vibratory movements covering atomic groups differing in scale [52], the advent of a sufficiently energetic thermal fluctuation and the rotation of any chain unit through a wide amplitude relative to a "neighb o u r " (T-G transition) will be precisely what is needed to set in motion a correlational chain segment (statistical segment) that is contiguous with the rotation axis. In this way the latter is isolated as a conformational mobility chain segment. Low amplitude torsional vibrations apparently determine indistinct or ill-defined relaxations at T<< T8 [92] and the relaxational background. The barrier height of B varies only slightly for the majority of flexible polymers, though here there are sure to be some exceptions. For instance, in the case of particularly flexible chains, e.g. polysiloxane chains, this barrier decreases to 2-4 k J/mole, while for highly rigid macromolecules the barrier will be higher. In the case of really rigid polymers, for which only limited vibrational processes are possible for chain fragments [53] that are much shorter than statistical segments the expression derived for Qa obviously does not hold. Moreover the results of an analysis of effective activation volumes for relaxation in the region of the fl transition Va also proved helpful in determining units of motion in fl transitions. The ana-

2752

¥. A. BERSHTEINand ¥. M. YEGOROV

lysis was based on a study of changes in the internal friction spectra (torsional vibrations at ~ 1 Hz) due to the application of static shear stress to polymers; the procedure involved and the method of analysis have been described in papers [54, 55]. It was found in the case of the five polymers investigated (PS, PMMA, PVC, PC and PE) that the parameter Fp~500-1500/~3 is commensurate with the volume of the Kuhn segment in the case of the studied polymers [21, 56], To substantiate the general nature of p relaxation processes in polymers it is helpful to proceed with direct evaluation of the kinetic unit (segment) for movements in bulk complimenting the results of thermoactivation analysis and the found relationships. The amount of such data currently available is small, though the first results are not at variance with the above concept. For instance, methods of fluorescence depolarization [57] and paramagnetic probe investigations [58] give for polyisoprene at T~> Tg a length of 4-5 monounits for the unit of movement, which is equal to the Kuhn segment (see Table). According to the IR spectra of methylmethacrylate oligomers at 20 ° (region of the fl transition in P M M A ) the "oscillatory segment" has a length of 6-8 units [59]; for P M M A S ~ 6 (see Table). It has been shown by molecular dynamics investigations that chemical crosslinking capable of a slight degree of deformation inhibits the motion of a chain fragment of the order of ten units [60]. The idea that the unit of motion in the chain is about the size of the Kuhn segment is supported by changes we observed in the longwave IR spectra of glasslike PS and P M M A under the influence of measured amounts of crosslinking of the chains: when the average distance between crosslinks becomes commensurate with or less than the Kuhn segment it was found that low frequency skeletaJ vibrations of the chains are suppressed [61]. Finally, a longwave ]R investigation recently yielded fairly direct information regarding the scale of movement in the fl transition: for a number of linear polymers the activation energies for ,8 relaxation proved to be close to the potential barriers of torsional skeletal vibrations of chain sections (v~ 200-260 c m - ~) similar in length to statistical segments in these polymers [92]: it was also found that intermolecular interactions play a determining role in the formation of potential barriers to such movements [93]. In our view segmental motion in the fl transition may best be described by a model of local movement in chains such as that proposed by Gotlib and Darinskii [62] which has subsequently been further developed in [63]. The model comprises an isolated T-G transition (single-barrier jump) with forced "adjustment" of a chain fragment contiguous with the rotation axis via rotation of units in this fragment through various angles decreasing in size as the distance from the transition site increases. There is little distortion of valency angles and bonds of the displaced chain fragment since the movement of each mono-unit relative to its neighbour is only slight (within the limits of torsional vibrations of low amplitude). The displacement of units apparently comes to an end at a distance away from the rotation axis which (according to our interpretation) is commensurate with a Kuhn segment; this is not at variance with computer calculations [63, 64]. The movement model described in papers [62, 63] is essentially based on a combination of rotational-isomeric (Vol'kenshtein) and torsional-vibratory (Bresler-Frenkel') mechanisms of conformational transition. Significance of the fl transition. The concept of the general nature of fl (T< T~)-relaxation in linear polymers involves a number of consequences, some of which are of practical importance. First of all one has to recognize that the onset of segmental motion with the participation of T-G transitions (conformational mobility of chains) in polymers appears not at the glass transition temperature, but locally, in sites of less dense packing, starting already in the region of the fl transition. Moreover parameters of the fl transition (potential barriers, size of units of movement) are directly connected with basic characteristics such as the thermodynamic rigidity of a chain (the value of S), the intermolecular interaction energy E¢ and the internal rotation barrier B. According to new data published in [20, 24, 26] the "large-scale" nature of the cooperative act of relaxation in the ~ transition (in the region of Tz) is likewise attributable not to the movement of unusually large fragments, but to intermolecularly correlated displacement of neighbouring segments whose size is the same as in fl transitions. This means that, starting at temperature T ~ T~, i.e. in a solid-state fltransition in the region of T~, and above T~, and in polymer melts, a determining

General mechanism of fl transition in polymers

2753

role is played by acts of movement appertaining to the fl transition. In o t h e r words, the length of the basic segment of movement varies slightly, and over the entire t e m p e r a t u r e r a n g e of conformational mobility is commensurate with the K u h n segment. It does not appear altogether remarkable that the relative stability of the segment of movement in polymers in the condensed state should be postulated. Experimental estimates of the extent to which the length of K u h n segments (i.e. in the unperturbed dimensions of chains in 0-solvents) ma~ vary with temperature (see tabulated data in paper [65]) show that such variations are generally quite small. For instance, for the interval zlT~ 200 K, which is approximately equal to the temperature range of conformational mobility of polymer chains, the K u h n segment may become shorter by ~ 1 0 - 3 0 ~ as the temperature rises, while sometimes it may be lengthened to the same extent (as in the case of PDMS), and in individual cases, e.g. for atactic PS and PM MA, it remains practicaJly constant. Of major importance in this connection are the data reported in [66, 67] regarding neutron scattering in polymers. These data show that at various temperatures (in the melt, in the glassy state) polymer macromolecules in the condensed state retain the same unperturbed for practically unperturbed) form as in 0-solutions. This means that intermolecular interactions do not here have any great influence on the flexibility that is a feature of isolated chains, and that the "rigidity element" viz a "correlational" chain fragment in a block polymer remains the same as in a dilute 0-solution. In view of what has just been said the line of investigation involving the statistical K u h n segment seems logical enough: the introduction of a theoretical concept of a segment into a freely jointed chain m o d e l - e x p e r i m e n t a l determination of the length of the segment in 0 - s o l u t i o n s - e v i d e n c e that it retains its role as a correlational fragment in the polymer b l o c k - discovery that the segmental length is commensurate with that of segments of conformational mobility of polymer chains governing the main (e and fl) relaxational transitions. F r o m a practical standpoint the value of the relations or'served for the fl transition lies in the fact that conformational dynamics of macromolecular skeletons, and the fl and c~transitions determine a large n u m b e r of physical properties, and processes, in polymers. For instance, in the stabilization of polymers (mechanical properties, enthalpy, volume, length and other factors)appropriate molecular regrouping takes place, as we have said, precisely in the temperature range (Ta-Tg) [16]. and moreover with activation energies Q~QI~ (e.g. see [261). The material discussed above shows that the viscous flow of polymers is governed by the same acts as those controlling fl relaxation; it is clear already from the work of Eyring that the length of the K u h n segment approximates to that of a "segment of flow" which, in the case of PE, contains 15-20 carbon atoms [46]. Free-radical and some other chemical processes occurring in solid polymers are determined not so much by the true interaction kinetics as by the conformational mobility of the chains; moreover their onset normally appears in the region of the fl transition, and the activation energy involved is Q z Q a [68-71]. Activation energies approximating to Qa are in many cases observed, for diffusion processes in polymers, e.g. for diffusion of gases (COz, CH4, 0 2 , N2): ~ 35-40 kJ/rnole for PE and PI, 60 k J/ /mole for PVAc, etc. [72]. Dielectric relaxation processes in polymers are determined by potential barriers of ~ and fl transitions determined by other methods in [1]. It seems that this could well have an important bearing on the plasticization of polymers if the latter is associated primarily wilh a complete or a partial transition from intermolecular cooperative motion (c~ relaxation) to a quasiindependent (fl relaxation) movement of the same segments [20, 94]. Finally, the forced elasticity of polymers is directly related to their relaxation spectra, and in particular to the e and fl transitions [34, 73-75]. According to paper [76] the activation energy of anelastic deformation of glasslike polymers Qa~

E~

ha, where n, is the n u m b er of m o n o m e r units

making up the activation volume of the process. It was found that at temperatures T ~ To n, ~- S [77], i.e. Qa~QB and the activation volume Va~ VB~ V~. It was also shown that potential barriers of Qa are fully in keeping with the activation energy spectrum for molecular motion (enthalpy relaxation) in the temperature range from the start of the region of fl relaxation to Ts [26]. In view of these

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and other data we surmise that the plasticity of solid polymers is realized via the “incorporation” ,of acts of mechanically activated /I and a transitions, i.e. during the displacement of chain sections comparable in size to the Kuhn segments. It is clear from the above discussion that the following equations, characterizing the Arrhenius B

(_Q%),

Q, z (0.3 k 0.05) E, S are satisfied for nonrigid linear polymers: v% lOI3 exp Q, kJ/molc + 15 kJ/mole, T,(K) = 7 V&q% v,. 0.25--0*0191og Y These formulas and related ones applicable to parameters of the a transition [20,24, 261 should lead to novel ideas and procedures connected with the prediction of polymer properties. This will involve use of calculated cohesive energies and lengths of statistical Kuhn segments measured in .&solutions. In the case of flexible polymers it can be said that the length of the Kuhn segment will be approximately 20 A. transition,

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Polymer Science U.S.S.R. Vol. 27, No. 11, pp. 2757-2764, 1985 Printed in Poland

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RELATIONS BETWEEN THE MAIN RELAXATION TRANSITIONS IN POLYMERS AND THE LENGTH OF SEGMENTS AND THE CHARACTER AND DEGREE OF COOPERATION IN MOLECULAR MOTION IN THE VICINITY OF T.* V. A . BERSHTEIN, V. M . YEGOROV a n d Y u . A . YEMEL'YANOv Ioffe Physicotechnical Institute, U.S.S.R. Academy of Sciences

(Received 19 August 1984) The DSC method has been used to investigate the glass transition (~ transition) in flexible linear polymers and oligomers, as well as relations between the ~t and fl transitions. The large-scale act of ct relaxation involves an intermolecularly correlated displacement of neighbouring segments of similar length, as in fl transitions and in polymer melts, i.e. approximating to K u h n segments. Non-Arrhenius values of thermoactivation parameters of ct transitions correspond at low frequencies to an act of correlated displacement of from three-to-five segments. AMONG the main relaxation processes in polymers are those of cooperative motion in the vicinity of glass transition temperatures Tg (~t transitions) and processes of fl ( T < Ts)-relaxation [1-3] determining polymer properties such as deformation, viscous flow, diffusion permeability, physical ageing, reaction kinetics and other factors. No clear-cut resolution of the problem of molecular assignment of ct and p transitions appears to have been achieved. For instance, there has until recently been no unified interpretation of the ,8 transition for different polymers [4]. In the case of flexible polymers the elementary act of ~t relaxation has frequently been associated with the movement of segments that are some tens of m o n o m e r units in length [3], although this scale has a flexible kinetic segment (Gaussian subchain) as observed by Kargin, Slonimskii and Rouse [1, 5] that is capable of highelastic deformation. This means that the segment of movement in the ~t transition must be a lot * Vysokomol. soyed. A27: No. 11, 2451-2457, 1985.