THEORI- O F VISCOCS DEFORJI.ITIOS IS POLTJIERs
(36) KIMBALL, G . E.. A S D GLISSSER, A , : J. Phys. Chem. 8, Sl5 (19401. (37) LISE,K . W,, A S I J ~ I C D O S A LH. D ,J.: J. .hi. Chem. Soc. 68, 1699 ('1946). (3s) IASGXIUIR,I . : J . .11ii. Clieni. Soc. 38, 2221 (1916). (39 I JIOELTTS-HCGHES, E . -1. : K i n e t i c s of Renctiom in Solution, 2rid edition. Clarentlon Press, Osford (1948". (40'1 SERSST, IT,: Z . physik. Chein. 47, $2 (1904). (41 I SOTEF, A . A , , ASL) WHITSET,W . R . : Z.physik. Chcni. 23, 639 (1897,. (42) XIIERWOOI), T. IC,, A S DT v O E R T Z , B. 13.: Trans. .11n.Irist. Chcm. Erigrs. 35, 517 (1939'. (43) SPASGESBERG, I-a.Chem. (L.S.1i.R.) 20, 1433 (lS4Gi. ( ~ S TL-, J C. >I., DAITS.H., A S D HOTTCL, €1. C . : Intl. Erig. Cliciu. 26. 749 (19341. (4:); \ V i i , u m x t s s , 31,:%. ph!.sik. Chein. 66, 445 (1909). (50) % D . ~ S O J - S S I ~., i.11.: ,J. Phys. Cheiii. (L-.S.S.It.) 20, SGI) (1946) (51) ~ I l I l l E R 3 L I S ,J. F.. ?.SD .\[cDos.\LD, 11. tJ,: ,J. Pliya. Colloid Clieiii, 51, 857 (19471,
-1 THEORI- OF T-ISCOI-S DEFORXITIOS IS POLTJIERS; THE Tv'C)LTXIE.ISD SI-RF-ICE COHESIOS,U ESERGI' OF THE R-WIC JIOT'ISG AiGGREGAITE'' D R lI0RET l i c d a l ii'car
(it
ch Laboratories, Rochesfo
4, S e i r * Y o r i
I ? c c e i i w l Jitlii 25, 1948
In n i m y phenomen:i involving rates, n quantity of energy. E , necemary to iitc or initiate a step may be apprownately entered in an eyponential term, ! ! I . and i+ deteiniined by I arying 2'. Thi.; paper is concerned particularly 1 it11 the interpretation ot thi* energy ~ n l u ein the viscoui flon of polymers, E,, incl 11 c>\perimentczll\- ubseired dependence lipon conditions of measurement I+ it111 tie noted presently, the values of E, are conbiderably higher than the 'nergicb4 oi single secondary boiids. In mo+t cases of viscous flo~l-,primary bonds lo not enter; the process inr-oh-es only secondary bonds, and hence, as has been \ell recognized by variouh iiriters in this field, a number of small, secondary )oncli are coordinated together in the unit or primary act of viscous flon-. Thus \ e come logically to the concept of a unit oi matter nhich i. displaced relative to tb neighLors, and vhicli i.; of considerable size in order to account for the large iumbei of small coordinated energies which milst add up t o tlie esperimentally Ill.er\ P d large vnluc l'he hn-ic~ problem. may be stated as follov b : (0) TYliat is the size and shape
1~t i \
-
1 Presented at t h e 113th 3Icetiiig of t h c \ ~ n e i i c ~ C'heiiiical ii Societv, Chicago, Illinois, pril, 1948. 2 C'omiiiunicntion S o 1209 from the Iiodak Rese:trcli I A o r s i t o r i e s
si0
&rI1\\TlO\ L\b H G >
POLHJEX
__
1
\IFTROD \ \ D RE\l\RhS
POL,,LLR
I
~
kco
?I10
Polyniet hy1 methacrylate
72
Frequcncy of Rubber maxiniuiii 1 I loss
Pol>methyl m e t h - ' acrylate 105 plasticizer
5b
Frequency
+
I nict\i
Polymethvl meth-l :tcr>late 207 plasticizer
3s
of Rubber
mum
loss
I
Frequcricy of inas i niu in loss
+
E'rccpencj of Rubbcr lil:~\il~lunl I loss
52
I I
Pol y inc t h y 1 niethacrylatc
I
1
23-46
,'
Polymethyl niethacrylatc
122
23-50
Polystyiene
Yield point;ll I I a i d lubber dependent I o n teniptv:r-' I1 tuir
I
d o \ \ from
1 Rubber late\
l-ielcl point ,I' Itubber tiepcndent on tempera-'1
I
39
Polyvinyl s c e t a t c Polyvinyl acetate
1
I
io-80
Itclaxatiori o f ' ~ o r n i v a r stress
4
1
1 chloride (plasticized) .
Tield point; dependent on temperature
I 5
Yield point
571
THEORY OF V I S C O W DEFORMATIOX IS POLYMERS
of the moving unit? ( b ) What secondary bonds link this unit to neighbors, adding up to the observed flow activation energy? (c) What is the law connecting viscosity with such structure, in order that such phenomena as non-Sewtonian flon- may be accounted for? Table 1 is a compilation of activation energies of viscous flow, showing the range of values and also the marked deviations n-hich O C C L I ~ . ’ I t is to be noted that the values obtained depend upon the type ofmeasurement chosen, and also that in several cases the authors report a dependence upon the temperature range chosen. I t is apparent that E,. cannot he regarded as having the qignificance and basic meaning which is given, for eyample, to dissociation TABLE 2 SecondarTj bond energies” POLYMEQ
YOLAX COHESION OF SECTIOS OF CHAIA’ TO IOT&OATEEB
1
A T T R ~ C T I X Gmaups
-1
Pol yet hylene Polyisohutylene Rubber Pol 5-c hloroprene Polyvinyl chloride P o l y v i n ~ lacetate Polyvinyl alcohol Polystyrene Cellulose Polvnniides
CII? CH:. CH:. CHz, CH=CzHz CHz, CH-CCI CH2. CHCl CH?, COOCB3 CH,, CHOH CH,, CsHj OH, 0 CH?, (C0SH)II
, I
I I
,
4.
kcai./mole
1.0 1.2 1.3 1.6 2.6 3.2 4.2 4.0 6.2 5.8
* From H. M a r k : I n d . Eng. Chem. 34, 1315 (1922). energies of primary honcL. The later development \vi11 bring out the more txrsic factors in EL.. .Titst as there is a well-defined activation energy for the formationof a primary bond, so also n-e may speak of the activation or dissociation energy of a secondary tjond. To illustrate the size of secondary bond energiec. table 2 presents c.al(wlatiuns hy l l a r k (19) on the cohesional enei Ijetn-een polymer chain$. It is seen that ,secondary bond energies are of the (~rtlerof 3 k c d . mole, smaller hy factors of 20 t>hanviscous flon- energies of these same polymcix. Considering first small-molecule liquids, in 71-hich there is little association, \\.e see that the moving unit becomes the molecule itself, and the activation energy of viscosity should drop to the value of the energy of‘ sec’ondary bonding ( J f ;L molecule t o its neighimrs;. \\-hen Jve deal ivith lo\v-vi.icusitj. liquids, such as ivuter and various alcohols, ive may consider a flon- unit to have an average size 1;irger than the single molecaiilc hecause of association, and E,. should :~lso rise 3 T h e authors mid references have heen omitted from t h e tnblrs i n t h e iritercst of hrevity, since the purpose is mercly t o establish :Ln o v ~ r - d lquditativ(, vie\\- r.:ithrr t h a n t o usc ipecific values in any qumtit:ttive w:iy.
572
D. R. MORET
above the \-alue of cohesional energy per single molecule. These general considerations are well illustrated by table 3, taken from an article of Grunberg and Sissan (15). I n the case of liquids, the flon unit is not a stable entity but under the influence of collisions is constantly changing size hy losing some molecules and adding others. -It the other extreme, for some high-melting materials, the flow unit is the crystallite and is quite stable in mass and shape. In between these tn-o extremes come our usual organic polymers, wherein we must consider the flow unit to be a J-olume of matter bounded by an envelope of weaker bondings. TABLE 3 Secondary bond energies* LIQUID
_____~______
Acetone.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . %-Octane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . n-Hexane Benzene Carbon tetrachloride Methyl chloride Methyl alcohol Ethyl alcohol Water n-Pr opyl a1c ohol
* From
~
kcal ./male
kcal./molc
1.68 1.9s 1.92 2.32 2.29 0.95 1 .os 1.3s 2.02 1 .til
1.67 2.04 1.82 2.52 2.51 1.59 2.48 3.29 4.15 4.39
Grunberg and Sissnn: S a t u r c 154, 146 (1914) bIZE OF THE FLOIV U S I T
There are several approaches vhirh may be used to get at the size of the flow unit, some of which are more direct than others: (1) The most direct method is that of observation n i t h the electron microscope. This presupposes adequate resolution a i d also that any boundaries or local qtructures observed are identifiable with the actual flonunits. (2) Estimated from the total f l o activation ~ energy, the energy per secondary bond, and the surface area to be associated n-ith each secondary bond. (3) Calculated from equations for viscosity which contain the volume of the f l o ~imit as a parameter. One such equation is derived and discussed in this paper. The viscosity equation of Eyring also contains a volume term which has a relation to the desired volume. (-2) Deduced from the dependence upon temperature of the frequency factor, F , in the equation
-1 _7
pe-ERT
57 3
THEORY O F VISCOUS DEFORMATIOX I N POLYMERS
( 5 ) Obtained from measurements of the scatter and absorption of supersonic vibrations. T o illustrate hon- ( 2 ) yields some information, assume that, fora given polymer, the activation energy of flon- is found t o be 80 kcal./mole. The secondary bonds we assume t o be of a type Jdiich has a value of 4 kcal.,/mole;then there are 20 of these taliing part in the process. If the polymer is crystalline, data obtained hy x-ray methods are a t kand on the spacing of groups responsible for these seconclary tioncis. Take 7 as 5n illustrative figure in this case. Then, with each bond we associate some 50 .I.? of surface. and for 20 bonds, 1000 \Yhat voli p e is associated v i t h this surface? ('onsidcr first n sphere. The sphere has 3000 A.3 volume. Rut if the unit is shaped like nn elongated rod, ivith the ratio of iengtli t o tlinmeter cqiial to 10, then tlic 1-olume is 1100 .\.."rrliis metliod of approach thus requires spccific knon-ledge of the energy T d i e aiicl spacing of the secondary bonds., ani1 a remonahle giie 1s t u s1i:ipe of the flu\\- unit. Even with these 1inI~no~vns evaluated, this metho vi11 give volumes much too lon.. hecause umed that eacah secondary bond is fully developed to neighhoring units. Hon-cver, the I-ery existence of a unit which can move x s nn entity shoivs that the surfacne hondings are not :it masirnun1 strength. Some of them may he bonded t u CJtlieK5 o n the same >.urface, t h i s sutisfying the furces h l i t not contrihuting t o the i e k n n c . e to floiv; others may he too fnr from a group on thc nest unit to c.xe1.t much nttractil-e forw. Tlie present development explicitly accounts for the clift'erent state of affairs at the surface of the flon- unit by use of a srirfaceenergy factor, Z ! which is s1ion.n to be basic t o the determination of the actiration c'riwrgy. E,,. In the nhovr considerations. it lias been taritly assumed that a hole or void \\-as accessible to the iinit nkicli is considered to rno1-e; this question of available holes is discussed later. The kinetic concept of polymer viscosity developed herein has its roots in the earlier work of Frenkel (12), TT-IIO expressed 1-iscosity in terms of molecular heat motions and an activation energy; he as also a u x e of the concept of holes in a lattice and their wandering from place to place. -41~0 u-orthy of note is the Ti-ork of Beckcr (21, which denls n-ith n small voliimc of matter as the element of flow, this small volume moving in discrete jumps i\-lienever the applied stress and the thermal mot'ions favorably combine to esceed thc binding force. These concepts have been vigorously developed by Eyring and his con-orkers (9, 13) into a well-unified and most important contribution t o modern theory. The basic ideas of Eyring lead to the final form.
.I.'
S being the stress applied and B containing a flon--unit volume. The hyperbolic sine function is a consequence of considering both forn-ard and backward jumps. This idea also appears in Prandtl's theory of elastic after2ffects (4). Prandtl's ideas have not been widely used because the physical node1 chosen is somen-hat artificial and difficult; it is for this correlation 11 ith nolecular processes, as n-e understand them, that Eyring's work is outstanding.
n.
574
K. MOREY
There is to be noted in the hyperbolic sine the quantity, B , which contains a volume. For ordinary non-associated liquids, this volume may be considered as approximately the volume of a molecule. For polymers, it becomes more questionable as to how closely it measures the actual yolume of the moving unit; the uncertainty arises in the interpretation of the ‘jump-distance,’ X, or the distance moved forward by the flow unit in each separately activated motion. I t has been assumed. in this theory, that A is a fixed length. Pmalln-ood (21) speculates that for rubber *,thedistance jumped is of the order of magnitude of a small multiple of the length betireen tn-o double bonds.” Rurleigh and Wakeham (5) speculate that for cellulose X is the identity period along the chain, 10.3
K.
FIG.1. Flon units a n d lowdensity regions of cellulosir-type polymer
I t is interesting that sonic calculations of this volume haye shon-n it to decrease rather rapidly with increasing stress ( 5 , 17). KISETIC DE1IIV.ITIOS
\\-hen it is recalled that the yiscosity of a simple liquid is in centipoises, whereas the values for a polymeric solid may he ab high as 10” poises, it is apparent that one is justified in leaI-ing behind the idea of a liquid state and discrete jumps in all directions. For a cellulosic-type polymer, it may be considered that the flowing unit is of good size and ne11 bound to neighhors. Furthermore, it is reasonable to assume that such units will move only u-ith the application of an external force. Figure 1 illustrates the sort of structure to he expected in polymers of the cellulosic type. We may expect large elongated aggregates of chains which cohere sufficiently ne11 to floir as one unit ; it is not necessary to answer the question as to whether these units are identical with structures giving rise to x-ray and optical effects. These regions fuse into one another with small
THEORY O F VISCOUS DEFORJIATIOS IS POLYMERS
575
secondary lioncls on the surface and with here and there commonly shared long chains. IT-e shall assume that holes are present, hut not the high-vacuum type of hole which oc~~111~ in simple liquids liy favorable collision ; R1 illustrates such a hole. Instead, it is more correct to ronsider, in associated polymeru, regions of low density into which an aggregate may move with little loss of' momentum. These regions may lie cr rossetl by flexible chains, as in R?,as long as little resistance is offered to the entering unit. Consider the material of figure 1 subjected to a tensile stress; then various rotations and translations of the flow units into available lon-density regions take place. The viscous flon- process may non- be thought of as consisting of three parts: ( I ) a breaking, by favorable kinetic collisions, of some of the secondary bonds ivhich hold a unit t o its neighbors; ( 2 ) a lireaking of the remaining bonds 1)y the applied force, resulting in a forlvard movement of the unit (among the f'ac*toi'sdetermining the velocity are the mass of the unit and the applied force); arid ( 3 ) a freezing of the unit to ne\\- neighbors after a time, r , the length of time of motion, T , tieing determined by energy content. So\\-, take as a representative sample of the ultimate floiv process most likely to occur the mol-ement of -42 (figure 1 ) into the void shown. TT'e may consider - 1 2 restrained to its old position 13)- forces t o &-l1 antl and urged to its new position hy forces transmitted through B, and B2. The process is thus one of shear. Let the external tensile stress be S dynes,'cm.'; then the maximum shear force applied on A 2 is Sa,'27r, a being the surface area of &d2which is considered to be a long cylinder in shape. In order that a permanent displacement he given t o - 4 2 , Sa ,277 must' exceed the restoring force, holding the unit in place. Represent by E ihe summation over a molar surface of - 4 2 of the secondary bond energies which hold - 4 2 in place. Furthermore, denote by E l the thermal energy in the surface bond system. Then the restoring force holding the unit in place is given by the derivative of this net energy, d ( E - E t ) ,d.1:. This restoring force \vi11 not be linear nit11 distance, but it \\-ill have some average value, v-hich n-ill be proportional to E - Et, denoted by 1 6 x ( E - E ! ) . Son-, for movement of the unit to occiir, the thermal energy must exceed some critical value; this critical value, E", is that value a t which the shear force on theunit just equals the restoring force, thus:
E;
=
E
- 2.3 X lO"6Sa cal. mole.
(1)
The nest step is to calculate the distance moI-etl f'orivard 11y the unit. and instead of assuming this distance, A, to lie fixed, l y e here consider that the unit may readily floiv past several small potential minima antl that it stops onl!. \\.hen its i u r f a c ~groups have lost 01s thermal transfer) cnough energy so that the kinetic energy of translation of the unit may he taken up in nen- bondings. This ,rings up the consideration of the lifetime of an excited thermal state, or the :ime requiiwl foi, a given quantity of thermal energy stored in various groups u c h as CH,, SH,, OH. etc., to he transferred away by rotational and vibrational ransfers to neighboring groups (which may he cwnsideretl to be those in the nterior of the flow unit as well as those 011 neighbor units).
576
D. R. MOREY
It is reasonable to assume that the lifetime of the state of motion is proportional to the excess of energy over the critical value, thus 7
- E;)
= +(ET
(2)
so that the greater this excess energy, the longer is the time needed for it to be dissipated away before the unit can “refreeze” to new neighbors. Xow, to get the average value of T , we need the average amount of thermal energy contained in the molecules which form the bond system tying the unit to its neighbors. Denote by z the average thermal content of one of the m a n degrees of freedom in the surface bonding system. Then the value of :, reckoned from any arbitrary lon-er limit, E ( , , iec-
/a
‘‘
de
+
Z = Le-_ - eo RT
I,
e-(
de
R1
For the entire surface bond system. -
E,
=
\EO
f TtRT
i.7\
n being the number of degrees of freedom in the surface bond system. Son-. assign to this lower limit, :eo, the value of the critical energy, E:. Then 7 =
that is, we average
7
+nRT
only over c35ei of actual movement.
(4 )
The thermal content,
E t , can and does take on values belou- E“, but these sets of collisions which do yield values less than Ectare counted in by the exponential factor, c-“ R T , n-hen n-e compute the number of times per second that movement occurb. \\-e now proceed to calculate the distance covered in an average jump of the aggregate. To do this, we may awime ( a ) that the aggregate is continually accelerated over the whole lifetime f, the acceleration being given by Sa 2 ~ ~ 1 and 1 Jl being the mass of the aggregate This assumes that the shear force is caused by long-range forces which do not change much in intensity over the whole jump distance. In other ivords, the driving force is considered as a force field, analogous to a charged particle in a uniform electric field or a particle in a gravitational field. I t is difficult to picture the exact origin of such ,z shear force field and to base the unit process of motion on molecular movements and short-range forces. Therefore it may be assumed ( b ) that the aggregate, when freed, is mored fornard by the force, Sa/2a, acting only over a short distance (or a short time) and hence receives an impulse rather than a steady acceleration. The result of this impulse is a fixed velocity imparted to the unit, of magnitude +‘Sa/2nM, 4’ being an appropriate proportionality constant. The distance covered, in one jump process, is then for case ( a ) one-half the acceleration times the square of t,thus:
R? T‘ n2 A, = Sa+‘ ____-_ 4Tdl
(3
THEORY O F VISCOL-S D E F O R J I l T I O S I S POLI-MEW
57;
and for case ( b ) , the velocity times P, thus: Ab
=
Sa++'RTn 27rM
These derived espressions substantiate the assumption made by Beclier ( 2 ) , in his theory of plastic flow, that X and S are proportional. The over-all flon- velocity of the unit is the product of X and the number of times per second the process takes place. The fractional number of collisions (coordinated over the surface of the unit) which result in a value of E: or greater, and which thus result in motion, is given by e-E' R T , and the number per second of such activating collisions is obtained by multiplying by the Eyring frequency factor, l:T,h. The f l o ~velocity of the unit [for case ( a ) ] is given 11y
The flon velocity per unit length and the viscosity are obtained by considering the length which enters into this velocity. S o n , if each f l o ~ unit moved, then the velocity would be divided by its length, I , but n-ith the concept of available holes into which the units move, not every unit is free to move a t one time, as seen in figure 1, The eflective length for calculating viscosity is then PI, where P is the number of units between holes in the direction of the movement. Son-, assuming Poisson's ratio = 0.3, the viscosity, 7. iJ given by:
'
1/3 tensile stress = rate of elongation per unit length
Substituting the value of E? from equation 1 and lumping some of the constants into B', n-e obtain:
6, which is half the distance betn-een potential minima (for aggregate displacement), 11 ill be about lo-' cm. To reduce equation 7 further. consider the aggregate to have the ideal shape of a long cylinder n-hose length, I , is ten times it> diameter. (Such a shape is reasonable for cellulosic and vinyl derivatives; for rubber and globular protein materials, the shape factor will be different .) Ken-. let 1' be the volume of the aggregate; and with the chosen Qhapefactor, I , Jf ancl (I may be put in terms of 1'. The density of the polymer, thu2 introduced, i y absorbed into B. I t is further considered that 1 2 , the number of degrees of freedom in the aggregate surface-bonding qystem, i h proportional to itc. surface area, u . Finally,
The choice of exponents indicated represents the choice between impulbe ancl force-field type of aggregate step-movement.
578
D. R. JIOREY
mscowrr Examining eqiiation 8, it is seen that an esponential function of stress and non-Seu-tonian viscosity is the natural state of affairs rather than an anomalous exception. This conclusion has already tieen reached from other derivations, as NOK-SEWTOBIAS
Torque ( D y n e C m r l
FIG.2. Non-Newtonian viscosity of polyisohutplene. Datu of Ferry sud Parks (Physic3 6, 359 (1935)).
THEORY O F VISCOUS DEFORMATION I N POLYMERS
579
Now, according to equation 8, a t constant temperature log p is a linear function of the stress, and from the slope of such a line may be obtained the volume, T-, of the moving unit. However. experiment shows that in a number of cases, one of which is illustrated in figure 4, this linear relation is not found. These facts d o not mean that the basic picture is in error; they arise from two implicit assumptions u-hich have attached themselves t o the development, and, like harnacle., milst he scraped off. First, the size of the flow unit is not t o be regarded a. fixed, but may indeed be a function of stress. Second, the energy value, E . is also dependent upon 1- and hence upon 8 , as will lie shon-n later. Bp~makingthe assumption that the influence of stress on E is less than that upon
FIL;.4. 1'0lyvi11~-l :icct:trc.
J1.y
-
120,000; tcrisilc stress npplietl in dry air coiltiition
the term containing "S. :I roiigh h u t simple means is a t hand for romputing T-. il'his assumption is seen t o I)P reasonable for the data of this paper, because of the similarity in slopes of the curves in figure 5.) Applying R tangent to the log 7 o.?. -desired stress, from two points on this tangent,
[--value- 11y this method :I!e probably in Ycriou- error for -mall Ytresze*, hiit I)(>'om? hrttei at large value. of Somc. calculations of 1- from thc data of figure 1 are given i n table 4, along with othei value'. from the literatim. It is -ecn hat. u i t h increasing \trwi. it iz predicted. h t h hy this method and hy the l a c e t a t e (Cklva 2 5 )
Polyvinyl ncct'tti., iiiediuni I (molecular-ncight data of J Scheele. I i o l l o ~ t l - % 103, 1 11043)) I
'
50 50
10 50
2
Tuiigsteri (siiiglc c r y s t a l ) . . . . . .
25
!I600
3
l*:spoiieiitial foriii tlerivctl Iiy 11. Becker il'hysik. %. 26, 923 (1025.
ment of parts of the specimen takes place, these portions tieing deterniiried 1 ) ~ the distribution of u-eali bonding+. Thus, the flow unit decreases in size. .i('TIV.\TIOS
E S E R G T O F FLOJI- .IS I F T X C T I O S OF TE1\IPER.\TURI.:
Since the actiyation energy and the 7-olume of the flax unit thus muj' depend npon stress, it becomes pertinent to inquire if, a t constant stress, temperature i,s also a factor in determining the flon- activation energy and the f l o unit ~ size. It u-ill first be shown that' such is experimentally the case, and next that sucli influences of stress and temperature h a w a simple explanation in terms o f the concepts of viscosity just developed.
THEORY O F VISCOUS DEFORMhTIOS IN POLYMERS
58 1
Figures 5 and 10 present striking examples of how much a log viscosity vs. 1/T plot can deviate from linearity, and hence indicate a variable flow activation energy. Table 1 also noted similar cases. Ewe11 (8) has collected in graphical form (his figure 8) similar evidence on a number of different materials, pointing out that they are “associated” in the liquid state. Fox and Flory (11) 5how hon., for molten polyisobutylene and polystyrene, the flow activation energy drops marltedlv n-ith increasing temperature. Over twenty year5 ago, Dunn (6) noted such deviations for associated liquids and stated that “a double or associated molecule will require a larger amount of energy than a simple molecule. Consequently, if the degree of association decreases with increasing temperature, the average 1-ahe of this heat of activation n ill decrease . . .” Still other example- coiiltl he gi\en, besides those alw mentioned in the introduction. The concept of a flon unit of variable size is thus by no means novel, but has been 1ecopiiizccl 11 idely, by numeroub 11orlters.” -1quantitative understanding of the matter all that remains as a nen- contribution. THL
z
FLCTOR O F SURF.ICE ENERGY
In t h e pieT-ious derixation E has heen defined as the total energy, over the It -hould, there-
nio1,ii .iiii,ic.e of a cho-en unit, of tionding t o neighboiing unit-
foic, l w piopoitionnl t o thi, wiface itrea, antl -o \\e may writt’
E
=
G X
Za = b X (iX 10?3ZT72
foi tht> -Ii,ipc factor of length = 10 diametei3. and equation 8 bcconic.
n.here Z i d n surface-energy factor; it is a measure of the density and intensity of nttrac.tii-e centers per. square centimeter of flon--unit surface, antl thus depends upon t h r polymer and the type and spacing of lateral groups along the chain. 3loreo~.ei.,it ineastires lion- effectively the possible bonds are developed, for, in distorted :incl amorphous areas, \\.hich must certainly lie the case bet\\-een flo\\units, the i m d s are not all in a state of minimum potential energy; not all are at 3 In :ill of t h c cases cited : i h v e 3t h e activ:itioii energy is found t o decrease with incrcasilig te1nper:iturc. T h e opposite c>Hcct is verj. uriusuitl, b u t figure 5 illustrates such a case. I’hcre 1i:is reccritly appeared :Lstud>- of t h e viscous flou- process in metals, hy Eyririg> Fredrieisoii, :tiid 1IcLacli:Lri (10). This paper presents :L iiuniber of new :Lpproaches t o the ion. of niet:ks, particularly in t h e iritroductio~iof random density fluctuations (of thcrnial )rigin: 3s n c c c s s x y for 1oc:il fion. processes. Some csperiincntal d a t a a r c given showing hat t h c iioiv activation energy is itself dependent upon teniperature, a1it1in fact increases )roportio~i:~lly with the absolute teniperatnrc>. Thc esplanatiori of this is also related t o :m tl o m d ensi t.y fluc t u it t i 011s. 111 vieiv of t h e results just indicat,ed on polyviiiyl acet’atc, of an increase of activation nergy n.ith teniperature, it may be asked if similar theory might not apply here. However, he niagiiitucie of the change is so much gre:iter in t h e case of dry polyvinyl itcctate (50 CUI. t o 200 k e d . i n 20’C.) t h a t it appears t o be outside any possible derisity fluctuation ffect, a n d its esplnnntion is thus t o be sought in terms of density of bond developnierit a n d he Z factor.
582
D. K . 3lOIZEY
a normal distance 0 1 ' angle, :tnd some arc' not cle~-elopetIat :ill, and 2 thus depends upon the order-tlisordw ( m d i t i o x at the interface.; ; iipon how the polymer wax laid down from solvent or melt : :ind holy it \vas drafted. In general, \\-e might expect Z to decreabe : ~ s1- licc'omes largel,, for larger giuups also mean larger spaces between them. A s u.il1 he .seen, Z ('an aluo depend tipon the temperntiire, and can act independently of 1- in this respect. Figure 5 shoivs log 7 ' ' ~plot ted I ' C . 1 7' ut fixed stre.;$eG. I he most striking thing is the m:vkecl deviat ion from linearity, and ivith a cwvature opposite t o r .
TEILPER.4TC RE
"C .
19.5 26.3 32.3 38.;
that orciinarily i o i i n ~ ~ ~. i - e nthe use of a Yi tenn clue.< not rem0i.e it. ' ~ i c i activation energy, from the slopes, increases flum 50 k ( d . at 13°C. to 200 lical. at 33OC., in the face of a 1- which (see table 3 ) is clecwasing. Here is ;I case of a rising 2-value, ivhich is not tinespectecl. T1ics;c (lata isefer to perfecatly dry material, in which c e thew is no oppoi~unityioi, plasticizing action. not even by moistiire. Thii,~, it is only Ivith an increase of' tcmperaturr that the material is able t o adjust t o lietter internal contact aftel, clnc~li shearing process. T h e process is similar to rcwystallization after a liqiiid has iwen quickly cooled t o an :imorphous solid state. If the amorphous solid is kcpt at a lmv temperature, long times are required foi, migration and reorientation. ;It higher temperatures, the piwess of chain adjtistment is made easier, ant1 tlic %-value rises. This is a
plienoiiienon \re11 linoivn in the technology of metal alloys and the formation of crystal phases therein. Thus, the activation energy for flu\\- is a quantity whic.11 is less fundamental than I’ and Z in determining viscous behavior, and the complete theory of viL~rosit)- is obtained only \\-hen, starting with a hasic form snch as eqiiation 9, there are superposed tlic additional relation.?. T‘
=
f ( T , ,S)
Z
=
G(Y,S )
101
-4t the present time, a theoretical evaluation of 10 seem3 far oft One approac.11 is to gather data siiitahle for their evaluation under knonn conditions; %-valiies may also he studied experimentally by blocking off yome of the attractiici groups. The importance oi the quantitieb I’ and Z in a theoretical elplanation of hou n plasticizer functions is evident. For dry polyvinyl acetate, Z has values vhirli are of the order of lo-‘ cal.:cm.’ VISCO5ITY THEORY . I S D THE: EFFECT O F HOLEb
The success of the concepts of Eyring and con-orkers in regard to the formation and r61e of holes in liquids brings up the question of the appearance oi such a concept in the dynamic theory. The idea of holes in liquids has been especially valuable in explaining the effect5 of pressiire on viscosity (14) and in relating t o heats of vaporization (20). In the discussioii on figure 1 it iras slio\\-n that the model for this dyn:imic* theory pictured, instead of holes, low-density regions, u h ~ c hare present ah a natural consequence of the formation of the solid rather than as formed by witable collision.. This latter proeel? noiild require too high a temperature for siicli large holes. I n the dynamic theory, the effect of number and size of holes appeal3 in the parameten lumped in the constant, B ; thus, B contains the factor, P , nhich expresses the number oi units in the shear direction lying het\\een surres>ive lo\\-density regions, P is thui proportional to the reciprocal of the 1 3 poxter of the number of “holes”; this means that the greater the niimljer of lon-density regions, the loner TI 111 br the T-iscosity TIIIXOTROPI
It is of interest to note that while equation 9 gives the vi,wosity only after a steady state has been reached, it does indicate the origin of thixotropic effects. 1hiis, ivhen I’ is a rapidly decreasing function of S,at the .start of a shear experiment, a numher of large unite must lie broken into smaller ones. lhis process requires the additional energy of creating neir internal siirfacae. This additional energy, over and alm\-e that used in the normal viscous floir process, must. be supplied by the external shear force. Thiis, there is an initial transient period during which the viscosity is higher than it is for the steady state when all flow units are reduced to the size consistent with the value of the shear stress. v .
7 ,
584
D. R. MOREP
The effect of orientation or parallelism of the f l o units ~ is not included in the discussions and derivations just given, but experimentally its effect is to increase q as deformation proceeds. This would be accounted for by an increase of the 2 factor as more and more secondary groups are made available for bonding of neighboring groups. CXPERIMESTAL TECHXIQUC I S 3ZE.ISCRISG VISCOSITY O F FILV
The importance of the stress value in flow experiments malies it desirable that esperiments be conductecl 0 0 that this p:irametcr may lie set at a knon-n value and
EXTESSIOS
U L V I ~ T I O NFROV C O S ~ T1x.r 1 STRESS
LSTESSIOS
U I : \ ' I . A T I ~ S FROU COSSTAST STRESS -- . .. ~
per
per
CLIII
0 J
10
20 30
0.0 +0.2
-0 1 -0.4
40
Cell1
-0.7 0.0
50 60
il.0
70
f3.3
-0.7
kept there, regardless of the actual amount of elongation in the specimen. The common experiment of hanging a weight upon a specimen does not give the desired results because :is the specimen stretches, its cross section diminishes and hence the force per unit area rises instead of staying constant. number of interesting mechanical devices have therefore been designed by various 11-orkers ( I ) to produce constant stress. For the experiments of this paper, conducted on polyvinyl acetate film strips, a device of good :tccuracy and convenience has been designed and constructed; this device is illustrated in figure G . The lever arm v-hich pivots around point P has attached a t its lon-er end both a weight and a flexible chain. The chain applies the stress t o the specimen as shon-n. As the specimen extends, it is seen
TEEORY OF TISCOCS D E F O R X I T I O X IK POLYMERS
587
TYPES O F SECOSD.iRT D O S D b IS POLYVISYL ACET-ITL:
F i g u ~ e1 shoned hou , tor polyvinyl acetate in a moisture-free condition, the viscosity dropped Jvith increasing stress. I n this dry state, we may consider internal cohesion to be due to : (1) chains jointly shared by t n o or more crystalline regions; this sort of bonding will be unaffected by moisture and furnishes a skeletal structure ; (2)water-sensitive bonds of the hydrogen-bond type ; ( 3 ) dispersion or induced dipole type; these too will be broken in a v-ater-sn.ollen sample; and (4) dipole bonds, Tvhich, for polyvinyl acetate, will be neglected. Figure 9 shows the dependence of viscosity upon stress for polyvinyl acetate swollen in water. I t
a"s- ' 0 9 - L IO'
.
?
n H,O
ot
193'C
I
10'-
\-__
H,O
-----in
01
290%
k----------inHeOa'365'c
10.-
1
,
I
'
is evident that thc more rigid the structure, and the higher the viscosity, the greater js the persistence of stress dependence. I n all curves, and indicated also by the data of Scheele (figure 3), n-ith increasing stress, the curvature decreases. This s h o w that the flon- units are being broken down to a minimum size. The data of figures 4, 5 , 9, and 10 show that higher temperatures and increased moisture aid the breakdown process, so that Newtonian size-stability is attained with loxyer stresses.6 T-alues of I ' for the curves of figures 9 cannot be computed with the constant 1.9 X los, for the value of 6 u-ill be different for the flow unit of the n-et state. It would be expected that 6 v-ould be considerably larger. The large difference in actual viscosities, between figures 4 and 9, shows that most of the internal cohesion in polyvinyl acetate is of the hydrogen and disper6 For the high-temperature cases, ~ i t water, h i t is necessary to extrapolate limited data t o indicate this, as these conditions are a t the limit of applicittion of the apparatus of figure S
588
D . R . MOREY
sion type, the shared chain contributing only a small cohesion. I t is of interest to note also the marked difference in internal flon behavior tietween the dry and wet states, a5 evidenced by the activation energiea. Figure 10, comparable to figure 3 , now s h o w the usual type of curvature. The activation energy of the flow process, computed from the dope, non- decreases, as the temperature i h raised, to a value of bome 50 kcal. ’mole a t 35°C‘. I t nil1 be recalled that, in the dry state, the value actually increased with increasing temperature, and since the stress curves shov 1’ to be decreasing, as the temperature is raised, this result showed the effect of temperature on the Z factor. S o w , for the casc of polyvinyl acetate swollen n ith water, Z-values should he constant, since readjustment after a deformation can take place readily. (The 2-values will he also much reduced over the value for the dry state.) Therefore, the diminishing \-ohme, T’, should alone he responsible for changes in the otiserved activation energy, E .
YISL:.\L
oiwm-.inox
O F FLOIV USITS IS SOLIDS
It is apparent that tlire,c$ obseryations ivith visitile light \vi11 not slion- up such sinall st,ructures as the cxlculations indkatc; honw-cr, the inrreasing application of t,he electron microscope offers promise of :I direct study of small flon discontinuit’ies. Such studies have already tjwn made! pai,ticularly for metals. For example, a note by IGiehler and Seitz (18) deals u-ith ideas of the size and movement of dislocations in crystals a s (’:tilies of phstic: flo\\-. ;\11 electron micrograph of et,ched (-oppcrs l i o u ~blocks some 2000 -1.in tliametei,. Elec.tron micro$(:ope work on etched :iliirniiium, put)lished by Heidenreirh :tnd dhocliley (1 lj), indicates “tiitit thc. (flo\v) lmiinae arc ahoiit 200 thick and of rather indefinite extent . . . The initial slip occiirs in :I single ( 1 11) plane (in the slip direction) until L: tlisplac~ement,of :itwiit 2000 -1.is t.cached.” It is cluent~ion:hle,tiou-r.ver. as to ho\v far ive cari transfer resiilts on metals io organic po1yrnei.h. .\n example of diwrt ohhervat ion of strurtural discontinuities (although not, brought nborit hy shear) is given by Spurlin, llartin, and Terinent (23). Their elect.ron micrograph of cthylcellulose film s1ion.s small but marlied
THEORT O F TISCOES D C F O R l L i T I O S IS POLYSIEIIS
589
structural volumes. These authors also give a schematic diagram of small-scale structure resulting f i ~ ma. longer chain netn-ork enclosing a plasticizer-smallr1i:iin phase. -111 these examples off’er the possibility of exact iclentificatiun of a flow unit, Iwt it iz still only speculation that thc units or volumes olwrvecl are identical ivith flo\v vulnines and have no other strurtural significance. 1he volumes incliteated hy the electron microscope 11-ould seem to c o r i ~ p o n c lto the flon. unit iintlei. very loiv shear; the calciilatcd valuen for sizahle shcai~stresses fall hehiiv the observed sizes. r_l
Fl-AIlf . i R T
.In analysis of viswm flo\v based on the dynamics of paiticle bonding :mcl mo\-enient leads to the folloiving expression f o r the viscosity :
l h e cbhoiceot eiponcntb dependi On thc i’ange of i ~ l t e l n fo1c.e i ~ ~ fields. J7 is the average volume in cubic centimeter* of the material uliich flous as a unit in deformation ; 2 i5 a smiare-energy facto1 measuring the i n t e n d y of secondaiy hond development pcr qquare centimeter of surface of the flon iuiit ; and S i, the applied tensile stress in dynes per square rentimeter. I n addition to this basic form, it is necessary, in order to characterize viscosity c~ompletely.to recognize the exirtence of the functionb: I .
TZ
= =
f’(T, S ) G(T,S )
The non-Sewtonian character of viscosity is accounted for, and is intimately tied in ith the size of the f l o unit. ~ For polyvinyl acetate film. it is shown that 1‘ decreases with increasing temperature hut that 2 may, under certain nonplasticized conditions, actually bc increased by a temperature rise. This accounts for the interesting hehavior observed on dry polyvinyl acetate that the activation energy of flon increases rather than decreases with increasing temperature. -4device is illustrated which automatically keeps the stress constant over deformations up to 50 per cent. From the curves of visco*ity us. stress, values are calculated for the volume of the floiv unit for dry polyvinyl acetate, using the formula:
where S is the tensile st,ress in dynes. KEFEXEXCEY (1) (a) ASDRADE,E. S . D A C . : Proc. Phys. SOC.(London) 60,304(1948). (h) ~ I O R E D Y ., it.: Textile Rescarch J. 15, 267 (1945). (c) WARD,A. G., ASD AIARRIOTT, li. It.: J. Sci. Imtruiiients 25, l4T (1948). (2) BECKER,i t . : Physik. %. 26, 919 (1925). (3) BUCHDAHL, R . : J. Colloid Sci. 3, 93 (1948). (4) BURGERS, J. AI. : F i rst Report on T’iscosity a n d P l a s t i c i t y , Roy. Acad. Sei. Amsterdam 16, 41 (1935).
