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1 AD REFINEMENT OF THE WILLIAMS-LANDEL- FERRY EQUATION D. S. Sanditov, et al Foreign Technology Division Wright-Patterson Air Force Base, Ohio 6 August 1973 q) DISTRIBUTED BY: National Technical'Information Service U. S. DEPARTMENT OF COMMERCE 5285 Port Royal Road, Springfield Va f :!,,

2 FTD-1T FOREIGN TECHNOLOGY DIVISION REFINEMENT OF THE WILLIAM&,-LANDEL-FERRY EQUATION by D. S. Sanditov, G, M. Bartenev '. < IiK7 U i:,@ C RePrduced b, NATIONAL TECHNICAL INFORMATION SE Ii ~ 5 ~ fleirv/ice Approved for public release; distribution unlimited.

3 FTD-HT EDITED TRANSLATION FTD-HT REFINEM.ENT OF THE WILTLIAMS-LANDEL-FERRY EQUATION By: D. S. Sanditov, G. 14. Bartenev English pages: 7 Source: Vysokomolekulyarnye Soyedineniya, Seriya B. Kratkiye Soobshcheniya, Vol. 14, No. 32, 1972, pp Country of origin: USSR Translated by: John A. Miller Requester: AFRPL/XPI Approved, 17or public release; distribution unlimited., F THIS TRANSLATION IS A-RENDITION OF THE ORIGI. NAL FOREIGN TEXT-WITHOUT ANY ANALYTICAL OR EDITORIAL COMMENT. STATEMENTS OR THEORIES, PREPARED BY: ADVOCATED OR IMPLIED ARE THOSE OFTHE SOURCE AND DO NOT NECESSARILY REFLECT THEPOSITION TRANSLATION DIVISION OR OPINION OF THE FOREIGN TECHNOLOGY DI. FOREIGN TECHNOLOGY DIVISION VISION. WP.AFB, OHIO. FTD-HT I Dat5 Aug 1973

4 U. S. BOARD ON GEOGRAPHIC NAMES TRANSLITERATION SYSTEM Block Italic Transliteration Block Italic Transliteration Aa A a A, a P p Pp R, r B 6' E 6 B, b C c Cc S, s B. B B 8 V, v T T T m T, t r r r a G,g Y y Y y U, u n /7 a D, d 4' 0 F, f E e Be Ye, ye; E, e* X x X x Kh, kh )K W. Zh, zh U U Li Ts, ts 3s 3 Z, z 4 hi Ch, ch HH H U I, i 11W w 1w Sh, sh 1i R Y, y LUW X 1 U Shch, shch xk KK K, k b Z' -b J1 )7 L, 1 hl y, y M -M M A M, m bb b ' H H H N, n E, e Oo 0 0 0, 0 0 )0 10 Yu, yu 11 n 17l P, p q R % R Ya, ya * ye initially, after vowels, and after 'b, b; e elsewhere. W-hen written as L in Russian, transliterate as y! or. The use of diacritical marks is preferred, but such marks may be omitted when expediency dictates. FTD-HT i U

5 UNCLASSIFIED Security Claazr.fication DOCUMENT CONTROL.DATA - R &,D esequilr r ellssilicllen of title, body of abstract and Indexing annot.tion must be entered when the overall report Is clasatlled) IORIGINATING ACTIVITY (Corporate auth or) 2a. REPORT SECURITY CLASSIFICATION Foreign Technology Division t"i'uclassified 2b. GROUP Air-Force Systems Command Us S. 3, R RORT TITLC Air Force REFINEM.ENT OF THE WILLIAMS-LANDEIaFERRY EQUATION C DESCRIPTIVE NOTES (Type oft potlt and Incluelve dales) " Trans lation S AUTHOR IZ (First niar.., middle Initill, lasi name) D. S. Sandit v, G. M. Bartenev 6REPORT [. DAMTE 7e. TOTAL NO. OF PAGES 7b. HO. OF' REFS,RO.,,.T,,O. ihi. rrftd-ht Cb. OTHER REPORT NO(S (Any other nuirbee that may be asaigned this re pot) -RAN () 7-7 T ' CONTRACTIOR GRANT NO.. ORIGINATOR'S REPORT NUMBER(SI) d. I 0 DISTRIBUTION STATEMEI4T : Approved for public release; distribution unlimited. II. SUPPLEMF.NTARY NOTES " 12. SPONSORING MILITARY ACTIVITY _Wright-Patterson '13. ADSTRACT Foreign Technology Division AFB, Ohio : 20 FFOR DD, NOV, 1 j UNCLASSIFIED ISecurity Ciassification

6 FOLLOWING ARE THE CORRESPONDING RUSSIAN AND ENGLISH DESIGNATIONS OF THE TRIGONOMETRIC. FUNCTIONS Russian English sin Cos tg ctg sec cosec sh ch th cth sch cach sin Cos tan cot sec csc sinh cosh tanh coth sech csch arc sin arc cos sin- I cos -I arc tg tan- I arc ctg cot - 1 are see sec -I arc cosec csc - l arc sh sinh 1 arc ch cosh - I arc th tanh - I arc cth coth- I arc sch sech - I arc each csch - I rot ig curl log FTD-HT ii

7 REFINEMENT OF THE WILLIAMS-LANDEL- FERRY EQUATION D. S. Sandimov and G. M. Bar.tenev The equivalence of the influence of temperature and time on the relaxation properties of polymers was first discovered by Kobeko, Aleksandrov, and Lazurkin [1, 2], and then confirmed by vast experimental data in the works by Linderman, Tobolsky, Ferry, et al. [3, 4,]. In polymers, physical relaxation involves two basic processes: 1) orientation and shifting of the free segments of the chain molecules, occurring very rapidly, with relaxation times of s at 200; 2) rearrangement of the elements of the supermolecular structure, accurrhng slowly, with relaxation 4 times within the limits s. In particular, viscous flow of linear polymers involves this second process [5, 6]. These two processes pertain, respectively, to the rapid and slow stages of physical relaxation. The process of the slow relaxation stage is found in the highly-elastic state, and can be approximately described by a discrete relaxation-time spectrum [5]. The process of the rapid stage of physical relaxation, playing the basic role in the glass transition of polymers,, can be quite accurately described by the averaged relaxatior time along [2, 11]. FTD-HTz

8 Table. Values of the constants of equations (i).and (5), characterizing the temperature dependence at(t) for various polymer. systems. Amorohous substance T 0,K a2 KIa 1 " a 2,iia ( c a OrGanic polymers [4] Folyisobutylene Natural rubber Polyhexane , Polyurethane D. 86 Polymethyl acrylatf '4 Polyvinyl chloride-acetate Polyvinyl acetate Ipanic olymers and glass [8] Vitreous boric oxide Lead silicate glass Alkali-silicate glass Alkalji-silicate glass No Alkali-s:ilicate glass No Ferry et al. [4] showed that the temperature dependence of the mechanical and electrical properties of amorphous polymers (and also il;,-molectular amorphous substances) above the glass., transition temperature Tc, approximately in the range T , can be described by one empirical function at(t), which is bhe ratio of the rapid stage of physical relaxatior at temperature T to a certain reduction temperature T a x(t)/t(t As 0 at =CT/( 0).A the reduction temperature we usually select Tc, which is the characteristic temperature of amorphous substances. Since the temperature dependence of the viscoelastic properties can be represented by the unique function at(t), it FTD-HT

9 is naturally very important to carefully study the form of this function and its significance. At piesent, the so-called Williams- Landel-Ferry equation [4-] is widely used, to calculate at: T- -a(1 A where the dimensionless coefficient a 1, 40 is a practically universal constant for all amorphous substances (Fig. 1),. Con-.tant a* has the dimensions of temperature, and initially it was also considered to be universal and equal to 500. However, thi.s value is valid primarily for a class of substances for which Tc K (see the table),. For inorganic glass (T c K), which recently has been examined as polymer systems [71, the values of a 2 300o [8]. Thus, the value of a 2 depends on the nature of the substance. In addition, equation (1) does not take into consideration the dependence of T on the rate of cooling or heating q = dt/dt c [1, 7]. In this connection, equation (1) is not universal, although it is frequently designated, as such. The physidal significance of equation (1).has been discussed in many works [4]. the constants In particular, we recently showed [9-11] that at U,/kT 0. (2) a.. (k TIE) T,, (3) where U c is the activation energy for transfer of a c unit (atom, segment) from one equilibrium position to another at T E is the energy of local disintegration, or, in other words, the energy of formation of micro-cavities on the order of atomic or FTD-HT-,

10 segmental dimensions at T c. It was assumed that the fluctuational formation and, disappearance of the iicrocavities in'amorphous polymers above T can be associated with the change in conformation c (convolution or evolution) of polymer chains due to internal rotation, and also rotation of side chains about the carbon chain.. The value of E practically coincides with the potential barrier of hindered internal rotation in polymers (1"4 kcal/mole, and was proportional to the glass-transition temperature [.11, 12] This approximate equality is, valid for all studied.-noncrystalline solids (Fig. 2),. Fig. 1. Correlation between. constant a 1 of equation (1) t o and Tc for oxrganic and inorganic O'0 49 SW 6O IX polymers -(graph constructed per data by Ferry [4] and Bestul,C, 0( 0 ) Fig. 2. Dependence of Fig, 3. Dependence of con-, energy of ricrocdisinte-- stant a 2 of equation (]) on grtocn'~fi aiu T 0 for various polymer syst~ems. amorphous substances [1,,c 12]. Designations: = kcal/mole. afa/motl=

11 Since kt /E const according to relationship (41), dependence C (3) of coeficient a 2 on T c should be linear. Actually, the p-lot., within limits of experiment error, is a straight-line with a slope, as was to be expected-,, approximately equal to (Fig. 3) q: -T 0,32 In this regard we easily see that equation (1) can be rewritten in the form [11] T-T In a, -- t T - Tca, + ( 5),, ị where the coefficient a 0 :..., '0,68,(6 ao U. unlike a 2,. is dimensionless and' practically independent, of the type of polymer (table).. Thus we have obtained a more universal form of equation (I). However, in equation (5,), just as in (i),, Tc cannot be considered a constant since it depends on the rate of change in temperature per the familiar Bartenev-Luk'yanov equation [13, 141]. c, (T- c, (7) where C '1, 0.03 is a uhiversal constant, while C varies from "l -3 to 10 - deg for various amorphous substances. This, equation, proposed by one of the authors [13], is valid fp, r a broad range of organic and inorganic polymers [7]. connection between equations (1) and (7) The inter" has been studied in prior works [9-11]; in particular, it was shown [9] that C _ kt /U 1/a. c C- 1 FTD-HT-23_ j

12 Consequently,,we must show at What cooling rate T was deterc mined, or reduce all Values of T to some single s'tandard Value. C According to Ba.rtenev and Luk'yanov [14.1, as the "standard" - glass-transition temperature TcT we can arbitrarily use that 2 temperature for which relaxation time T 10 s and the viscosity is l. 3 poise. The glass-transition temperature determined by d-i"ilatometry at a cooling rate of 3 deg/min (0.05 deg/s) approximately corresponds to the standard temperature.. Considering that when q= q, 3 deg/min in,equation (7) constant C 1 is -l/1.04tc, it is easy to obtain a formula for the transition from T, measured at an arbitrarv cooling rate, to Tc,,easured at the standard CT rate q = 0.05 deg/min: T _ o\-l (8 T. TC( 7-0;031ig q / From this we see that a change in cooling rate by one order of magnitude leads to a change in'glass-transition temperature AT c = 003T c. For example, when T = K, the value AT, 230, i.e., the error will considerably exceed the usual experiment error. Considering dependence (8), equation (5) can be rewritten in stricter form: T -Y o / l -O O l IN a,.. a, - 0 'igq' -"(9) ' Let us note that in the dilatometry of polymers and glass, in many countries a cooling or heating rate close to the standard 3 deg/min is used; therefore, literature data for the glasstransition temperature actually refer to standard temperatures. Consequently, in practice we can use equation (5) in place of (9), considering that Tc CT T c in it. FTD-HT:

13 s hswr recently by Sloniskiy, Askadskiy, and Kit aygorodsk-iy' [I.]o te raacking coef Zicient fs-r riolecu'es of' arno-rohous p~olymers knavins tbe =-a diverse stcru--t-urec. is equal tb -0.68, ieit ezirlailaes 'ihthe value of' the constant a 0 1 I /E) in C 'eq-at-lan (5). The Packing ccef'1icier.' i- h atoo the naual vali~e a1' Ihe atoms mtsng he molecules t o th. Ie '~ the s4hsce -eined per its den-sity. Coniclusi ons We, -rcrase a r'efinement ci' the Williams-Landel-Ferry eauation f~ t -e ter-, era ture denendence of the relaxation time during the rpid st tge of ph-4sical relaxation of' polymers above the - ga~strasi~cnte. erature T. in. this equatixon we have con" S~ee the d enaenzce ci' T n the cooling rate; In place of c e ffiaj int a,,, hav-r:i ng the di..iension of t-emperature.and a functiony tfth ype of polymer, we have introduced a dimensionless., practio aliy universal constant a =.,l-a /T 0 ) ,scrw State?edagogicl- TsiueRcie im. V. T. I-enin 26' IMa-ch 1971 Burvat S t at4.e Pedagogical Instit-ute...m. D. Bandarov A RIEFIERENiCES No 6e r.o, E. B. N y n m l i rc j;iti, r. It. r y p e n i % lao;. Ab CCCP, cepixa qklzi',63, '; 11. If. N o 6 ei. o, Axop(mnexe ner-tecaa, 13;-no All CCCP, 195T2. 2A. IT. ka e;c a rip on, 10. C. 271-a3ypir,!H. Iexn. (bg3mar, 9, 1250, L A. T o G sx16c.in i-f, CaoucTBa ni crpymypa nonxtepoxi, uiag-xo rxiurrn, I. Ax;u. (1) op p i,. 1b131-ytipyrne cisoictb3 IIOIImepoa, 1113-po unocrp. anf., 19G3. r. n. T. A-, it e D, It. 31. jf a a i a, B1,xc-o3oetc. 6D(. A-12, r. '.%. B3IPT IeDx, 11. -!. t xia, H Bmcomowoei coet3., -92, 02.1, TP ap Px T.e; C'D CTpOCUHt2 Ni mesafiagecae c,-,o:ctba xweipr.ia~iqec1;1y CT010:.. B.A B psn 17i.hitechn. fler.,,32k 5, SpreteA.C aayioncxn, 1.P~mnia,1.A l Lt 110, H.' Yp. %(Irmzl 159, 069