Modulus of Natural Rubber Cross-Linked by Dicumyl Peroxide III. Some Molecular Interpretations, Possible Refinements of Theory, and Conclusions

The shear modulus G = 5.925 × 10−3(fp − 0.45)T+G* (Part I), its energy component G* = 0.0684 (fp − 0.45)+ 2.70 (Part II), and the number of effective suh-chains per unit volume ve= (G – G*)/RT are given detailed molecular consideration. G is given in Mdyn cm−2 for rubber cross-linked by adding p parts of dicumyl peroxide per hundred of rubber, and heating until a fraction f of the peroxide is decomposed. ve is found to be approximately twice the density of cross-links, after a correction for impurities and chain ends is made. It can not be computed as G/RT since only the entropy component of modulus is related to ve. The sub-chains for the most highly cross-linked rubbers studied had a molecular weight of about 575 g mol−1, corresponding to about 8 isoprene units. The modulus corresponding to no added cross-links is not zero. It is determined chiefiy by the energy component of the modulus; it does not arise from entanglements. The “front factor” is found to be unity. An extensive literature survey yields values of the quantity RTΨ(v2), where Ψ (v2) is the Flory- Rehner equation function of v2, the equilibrium volume fraction obtained by swelling the cross-linked rubber. RTψ (v2) is found to be greater than G – G* but not as large as G itself.


. Introduction
The fir st paper in thi s series Part I [1] I presented experimen tal data on th e c han ge of modulus with temperature and cross -linkin g for natural rubbe r cross-linked by dicumyl peroxide , The results we re presented in th e form of the following equation: linkin g deviated from th at give n by the equation, and is discussed in a later secti on of the prese nt paper. The second pape r Part II [2] applied thermodynamical conside rations to the experime ntal results to determine the relative importance of G* (= H (jp + B) + A )-the energy co mpone nt of the modulus-and (G -G * )-the entropy component. The experime ntal entropy component was then compared with that predicted by the statistical theory of rubber elasticity in an extremely simplified presentation.
The general concepts outlined in the books of Flory [3], Treloar [4], and Ferry [5] were utilized, largely using the form and symbolism employed in the book by Meares [6].
The present paper makes much more de tailed molecular interpretations of the results than the second paper and e xamines the conseque nces of several refin e ments which might be mad e in the application of the theory,

1, The Gel Point
The gel point or degree of cross-linking just neces-451 sary for the formation of a network has been determined experimentally in Parts I and II by noting the degree of cross-linking at which there is no change of modulus with temperature (i.e., aG/aT= 0). At lower degrees of cross-linking this derivative is negative, as normally observed for uncross-linked polymers [7][8][9].
At higher degrees it is positive, as is normal for crosslinked polymers.
The gel point is reached only when the cross-linking ha~ attained so me characteristic value. In the present study it was found experimentally that 0.45 phr of dicumyl peroxide must be added to attain the proper degree of cross·linking. Some of this represents dicumyl peroxide wasted by r eaction with impurities in the rubber, and the remainder is required to produce, on the average, one cross-link per rubber molecule. Our calculation s of the relative sizes of these components have been based on an assumption regarding the amount of impurities in the rubber. However, the total amount of added dicumyl peroxide required in our work is 0.45 phr regardless of the relative sizes of the components and regardless of the value of the constant H in eq (Ll) No attempt was made in Part II in the section deal· ing with the theory of rubber elasticity to ' predict a value for G*, the modulus of the rubber at the gel point. The theory was used to predict only G -G* , the increase of modulus caused by the addition of active sub·chains beyond this critical amount of cross· linking.
The modulus of the network at the gel point at all temperatures in the present study is found by direct experiment to be equal to A = 2.70 Mdyn cm -2 regardless of whether the constant H is zero, as in eq (1.2) of Part II; or 0.0684 Mdyn cm -2 (phr) -1 as in eq (Ll) of Part II (eq (Ll) of this paper).
The statement may be readily verified by an ex· amination of figure 2 of Part II. The modulus at the gel point is likewise independent of the value assumed for the dicumyl peroxide wasted by reaction with impuri· ties, although this is one of the factors determining the abscissa of the gel point.
It is quite surprising that many previous workers have generally neglected consideration of the gel point and its effects on modulus. Most authors, with certain notable exceptions [10][11][12][13][14][15][16][17][18] have even failed to mention its existence. Charlesby [12] has given one of the most extensive discussions of the subject.
The nonzero value of the modulus near the gel point has likewise not been clearly demonstrated previously, although the experimental evidence has been available.

Linking and Related Quantities
In Parts I and II, the amount of added cross-linking has been expressed in terms of the quantity fp, since that is the independent variable directly observed ex· perimentally. It is of course pertinent only to the sys· tern of natural rubber cross·linked by dicumyl peroxide. For comparison with other systems and other measured properties the cross·linking can be alternatively expressed in terms of other quantities also closely related to X, the number of added cross-links.
Utilizing the numerical values found in Part I for the constants, we shall give the expression for calculating each of these quantities in turn in terms of the quantity fp· For convenient reference all the numerical values used in the present paper are summarized in the Appendix. Some of the new expressions have a greater significance in terms of molecular considerations than those using fp. However, the values are dependent on the validity of the theory outlined in Part II and on the constants which have been determined from it. Some of the values of course apply only to the particular sample of rubber studied here. Unlike fp, the values considered here are not directly observable quantities.
One of the most fundamental of these quantiti es This quantity may also be weritten as (1/2) Vr l M;:~hem where Me ehem is the molecular weight of a sub·chain (i.e., between cross-links), as determined from stoichiometry. Thus  Aside from th e e nergy co mpon e nt of the modulu s , th e fundamental quantity governing th e physical properties of the cross-lin ked network is th e number of effectiv e s ub-c hain s. Thi s may be expresse d in term s of the calc ulated number of effec tive s ub-chain s per mol ec ule, (2X/N -2), give n in table 2 and calc ulated from eq (2.2. 1), furnishing the figures given in tabl e 1. However, for many purposes it is mos t conve ni e nt to ex press thi s dependent variable as V I' , th e number of moles of effectiv e s ub-chain s pe r unit volume, whi c h may be calc ulated from fp by ins e rtin g in eq (3 .8) of Part II th e valu es of the cons tants de te rmin ed in Part II. From thi s we conclude that in our rubber there are 9.36 /LmoJ of dangling ineffective loose ends in each cubic centimeter (eq uivale nt to one added crosslink per rubber molecule or 2 sub-chains per molecule) at all degrees of cross-linking. Figure 1 presents a graph (dashed line) of this equation and table 2 s hows values calculated from it. It can be seen that the line is displaced from the origin by an amount fp = 0.45 phr, corresponding to the sum of the shifts, 0.31 phr due to impurities in the rubber and 0.14 phr due to dangling loose ends of rubber molecules.
We find that V e is zero for fp = 0.45 phr, where one cross-link per molecule has been added. Above this point it increases linearly withfp.
The dotted line in figure 1 shows the res ult of omitting completely any correction for the end-effect. It re presents the behavior expected if the molec ular weight of the polymer had been assumed to be infinite.
For comparison with these values predicted by crosslinking considerations from simple theor y, the observed value of V e may be obtained directly experime ntally 453 206-2E7 OL - 76 -5 from the observations of the modulus increase (G -G*).
With the insertion of the constants from eqs (1.1) and (2.3) of Part II, eq (3.9) of Part II becomes The subscript G has been added temporarily at this point to emphasize that this Ve has been obtained experimentally from the observed values of (G -G*) and not from cross-linking considerations. This equation is represented by a continuous line in figure 1. The positive ordinate values are 6.7 percent higher than those given by eq (2.3.1). The discrepancy arises from the 6.7 percent difference in the observed and predicted values of the constant S, already mentioned in Part II and tentatively ascribed to the effect of entanglements. These lines illustrate graphically the extent of the agreement between the number of effective sub-chains calculated from the observed modulus and that calculated from the added cross-links. With the correction for wasted dicumyl peroxide and for the chain ends, the relation is surprisingly close to demonstrating the production of two effective sub-cbains for each cross-link formed over the range shown in figure 1. This was of course assumed in the derivation of /Je in eq (2.3.1) by crosslinking considerations, but this assumption has sometimes been questioned. The conclusion may also be phrased in slightly different terms. The equation of the continuous line in figure 1 in terms of the upper abscissa is obtained by · combining eqs (2.2.2) and (2.3.2) to give  Each mole of added cross-links thus leads to 2.134 moles of effective sub-chains. If 6.7 percent of these are ascribed to entanglements it may be concluded that our experiments show that each molecule of decomposed dicumyl peroxide gives rise directly to exactly one new cross-link and two new sub-chains.
While one is justified in calculating Ve by dividing (G-G*) (the entropy component of the modulus) by RT, as has just been done, no significant information regarding the number of effective sub-chains can be obtained from G or G* alone. The ratio of either of these single quantities to RT is not a proper measure of any kind of sub-chain density, since the component G* is not related to entropy. Unfortunately most previous authors have based calculations of physically effective sub-chain density on the ratio GI (RT). This can be valid only if G* is neglible compared with G. Table 1 of Part II shows that this is far from true in the present investigation. This point deserves special emphasis because of the confusion in the literature.
It is clear that eqs · (2.3.1) and (2.3.2) predict no variation of Ve itself with temperature. However its efficiency in producing an entropy component of the modulus is proportional to the temperature, falling to zero at 0 K, as is obvious from eq(2.3.2). The entropy component of modulus at a given temperature then bears a close relation to the number of moles of effective sub-chains per unit volume and is in fact equal to RT times this number. G*, on the other hand, is an energy component related to intramolecular or intermolecular forces. It is assumed that its magnitude does not vary with temperature, and its variation with crosslinking is found to be very small, arising only from the presence of the term involving the constant H in eq  In most of our work we prefer to follow the practice shown in figure 1, where the de pendent variable is V e, the number of moles of effectiv e sub-c hain s per unit volume, and wh ere th e ind e pend ent variable is th e cross-linking expressed in term s of fp for th e dicumyl peroxide sys te m or in te rms of X/A I V,. in ge ne ral. In this way one avoid s the reciprocal quantiti es and possible confusion involv e d in usin g Me pllys and Me c ll ell/.
A co mpari so n of th e respective values of Me c hern in th e last column of table 1 with those of Me I) h ys in the last column of ta bl e 2 s how s th e necessity for making a careful di stin c tion between these two quantities. The di stinction is always important conceptually and is quantitatively mos t signi ficant at low degrees of crosslinking. Th e di s tin c tion wou ld di sa ppear for a polymer of infinite molec ular weight as can be noted from eq (2 .3.4) .

Measurements of Equilibrium Swelling
As an alte rnative to experim e nts involvin g deformation of cross-linked polyme rs by mec hanical stress, many oth e r inves ti gators, [10][11][12][13][14][15][16] have measured th e s wellin g of the syste m by a liquid. The inte rnal stresses ari sin g from th e swellin g are balanced by th e reac tion of the e xte nded ne twork. Thi s is observable as an in c rease of volume, approac hing a limiting equilibrium valu e.
Flory and Re hne r [40][41][42] have utilized s tati s tical co nsiderations to de riv e a relation inte nded to pe rmit the calc ulation of th e apparent numbe r of effective subchains from the obse rved limitin g equilibrium swellin g. Alternativ ely thi s number can be e xpres sed as For reason s that will soon become e vide nt, the de pe nde nt variable in this sec tion will first be taken, not as V e, !Jut as (G -G*), th e e ntropy co mpone nt of th e modulu s at a tem pe rature of298.15 K. This variable mu s t be divided by th e factor RT to obtain V e itself.
In terms of mod ulus components, the n, th e Flory-Rehner equation ca n be writte n: whe re V2 is the meas ured volum e-frac tion of rubber in the swollen material , VI is th e molar volume of the swelling liquid, and JL is the Flory-Huggins interaction parameter c haracteristi c of the rubb er and swellin g liquid. It is normalJy a bout 0.4. We s hall de note th e right·hand me mber of th e equation as RT'I!( v~) for brevity.
Values of RT'I!(vJ, th e ri ght-hand me mber of eq (2.4.1), calculated from equilibrium swelling data re ported by fiv e diffe re nt observers are plotted as ex-perimental points in figure 2. The results of van der Hoff [21] , C hasse t and Thirion [23], Plazek [29], and Allen a nd co-wor kers [36] are reaso nably well re presented by th e dotted lin e, which has been drawn to represent th e m. Similar res ults (not plotted) were also obtained by Angere r [37]. Th e res ults of Tamura and Murakami [38] are co ns is te ntl y so mewh at below the dotte d line an d s how a somewhat s malle r c han ge with in crease of cross-linking.
Th e data plotte d in figure 2 are also typi cal of the res ults obtained by many oth er observ ers [1 3-16, 19 ,20 , 22, 24-28 , 30-35]. The latte r res ults are not plotted here in order to avoid complexity. However, th e points are generally found to li e near or below th e dotted lin e . In no case do they fall below the plotted results of T a mura and Murakami [38], whi c h are re produ ced here to typify the extre me case. Most of th e differe nces among th ese res ults can probably be explained in te rm s of diffe re n ces in th e molec ular weight of th e rubber, its impurities, or the conditions of c ure and swelling.
The co ntinuou s line in fi gure 2 represe nts a plot of (C-C*) at 298.15 K. in accordance with eq (2.3.2). It represe nts the e ntropy co mpone nt obtaine d in part 1 of th e prese nt study, a nd differs from the continuous lin e in fi gure 1 only by th e constant factor RT in th e scale of ordin ates . As already de mon s trated in fi gure 1 the valu es of Ve calc ulated [rom cross-linkin g consid e ra tion s differ onl y sli ghtly from those corres pondin g to thi s co ntinu ous lin e. Th e negative interce pt can be ascribed to th e prese nce of c hai n e nd s and impurities in the rubber, wh ile th e sli ghtly greate r slope can be ascribed to the presence of e nta ngle me nts .
It is clear from fi gure 2 that th e values of RT'I! (V2), th e ri ght-hand m e mbe r of eq (2 .4.1), obtain ed from s welling meas ure me nts are sys te matically s ub stantially larger than the valu es of (G -G*) . In other word s we find th at th e e ntropy compon e nt of th e modulu s effec tive in limiting swellin g is larger than th e e ntropy component measured by the other methods.
The co nclu sion th e n is th at the number of sub-c hain s limiting swellin g is larger than the numbe r calculated from cross-linking considerations or its approximate equivalent number effective in the mechanical measurements of the present work.
Furthermore the slope of the dotted line in figure 2 representing the results of the five investigations [21 , 23, 29, 36, 37J is about 16 percent greater than that of the continuous line. The comparable figures obtained from the observations of Meissner [14], Manik and Bane rjee [31], Mullins [20], and Redding and Smith [32] are 10, 7, 6 and 3 percent re spectively. The data of Mason [25] and Tamura and Murakami [38] s how e ve n lower slopes, nearly the same as that of the continuous lin e. Added cross-links are appare ntly s li ghtly more effec tive in increasing th e e ntropy compone nt of th e swelling modulus than they are in in cr easin g the corres pondin g e ntropy compon ent of th e mec hanic al modulus at a give n temper ature.
These two di scre pan cies of ordinate values and slopes of th e lines in fi gure 2 are not du e to differe nces in spec imens, differences in calc ulating ab scissa values, or difference in the methods used in meas uring the modulus by mechanical means , since values of G calculated from stress relaxation observations of Chasset and Thirion [23J, from c reep observation s of Dickie and Ferry [30] or from torsion pe ndulum measurements of Plazek [29] on specimen s represe nting the same samples as those used in their swelling experiments are in excellent agreement with our values_ This comparison is shown in figure 8 of Part I. Additional experimental points in good agreement with this figure have been obtained more recently by Gent and Kuan [43] in torsional experiments and by Tamura and Murakami [38] by linear extension.
For comparative purposes figure 2 shows also as ordinate a plot (dash ed line) of G, the sum of the e nergy and entropy compone nts of the mec hanical modulus at 298.15 K, as obtained earlier by inde ntati on meas urements. This, of co urse, corresponds to eq (1.1 ) and to one of the lines in figure 2 of Part II. Its slope is about 3.8 percent greater than that of the plot of (G -G*) because of the presence of the term containing the  c:J Chasset and Thirion [23] or Plazd- [29] 8 Allen et aJ. [36]. s hown never exceeds the total modulus obtained by mechanical meas urements . It should be noted that the re is no justification for drawing a line in figure 1 to correspond to the line G in figure 2, since th e energy component of G does not arise from any definite number of cross-links, as e mphasized in section 2.3. This same discrepanc y of ordinates has already been pointed out by other authors. Dickie and Ferry [30] present a graph (their fig. 2) showing that the equilibrium compliance observed in their creep measurements was smaller by a factor of 0.71 than that calc ulated by eq (2.4.1) from Chasset and Thirion's swelling measureme nts [23]. In terms of the quantities we have used, this m ean s that the directly-observed modulus G was found to be greater then RJ'Ilt(vJ by a bout the same factor as we note in the region where jp is between 1 and 2 phr. Similar results for natural rubber and polybutadiene in this range were reported by Shen, Chen, Gebhard, and Cirlin [35] and for styrene-butadiene rubber a nd butyl rubber by Nielsen [44]. Murakami and Hsiue [39] extend ed the observations on natural rubber to higher degrees of cross-linking than the other observers. They confirmed previous work at low values, but in addition they found that 'l' (V2) from swelling measurements (which they denoted as ns) was less than GIRT (which th ey denoted as nJ/) only as long as the value of GIRT was less than 420 JLmol cm -3 corresponding to a G value of 10. In contrast with the pre ce nt paper, mo st pre vious publications have neglected to include G* in the Flory-Rehner equation and thus have considered its left-hand member to be s imply G, the sum of the entropy and energy components. If this should be correct, RT'l'(vJ ought to be compared with G rather than with G -G*, and it would not be necessary to conclude that there were more cro ss -links effective in limiting swelling than those calculated by the other methods. From figure 2 one would then co nclude that RJ'Ilt(V2) could now be ta ke n as a total modulus includin g an e nergy compone nt which is roughly only about half that effectiv e in the mechanical measurements. If it is assume d that in the swollen system inter-chain forces are very greatly reduced, while the intra-chain forces are not much affected, a tentative guess could be made that these types of forces would be approximately equal in the unswollen rubber. On the other hand, most previous workers consider that the intra-c hain force s are strongly predominant. However, the present work yields no information about these forces if the left-hand member of eq (2.4.1) is take n as (G -G*), as we hav e done.

. 1. Impurities and Sol Content
Specific impurities in the rubbe r whi c h would react with the dicumyl peroxide rend e rin g some of it unavailable for cross-linking have bee n di sc usse d in sections 3.2 and 4 of Part II. In the absence of direct measurements, the equivale nt amount of these impurities in pal e c re pe rubbe r was take n as 0.31 phr, as fo und in swelling meas urem e nts made by van de r Hoff [21]. Similar meas ureme nts on sa mpl es of several extracted and unextracted natural rubbe rs by Bristow, Moore, and Russ~ll r281 and other meas urements re ported by Bris tow [27] yielded values rangin g from 0.2 to 0.45 phr, in approximate agreement with van der Hoffs figure. This number would be expected to vary with the type of rubber studied .
Degradation of th e rubber is muc h more likely with extracted samples , sin ce extraction re moves the natural antioxidants. Consequ e ntly in most in sta nce preference s hould be give n to res ults obtained with un extracted sa mples, makin g the necessary correction s.
In general, all impurities in the rubbe r, both reactive and nonreactive, would also act as dilu e nts, for whic h an additional correction mi ght be made. A s imilar correction mi ght be made for impurities in the dicumyl peroxid e itself. In eac h case the mass should be multiplied by th e corres ponding purity.
The purity of th e pale cre pe rubber may be es timated as normally about 93 perce nt. The purity of the " recrystallized" di c umyl peroxide was not measured here , but values near 95 percent have been reported Therefore the correction factor would involve only the ratio of the purities. It would be unity if the purities of the rubber and dicumyl peroxide s hould be the same.
Since thes e purities probably were nearly th e same in the present work, no correcti on for diluent effect was made.
The reaction products of the deco mposition of the dicumyl peroxide are a-a'-dimethyl be nzyl alcohol, acetophenone, and methane . It is expec ted that a portion of th e first two products will remain as a residu e in the rubbe r. No significant anomalies due to th e prese nce of th ese residu es were note d in the experimental portion of th e present study, and no account was taken of th e m in th e simple theory outlined he re. Howeve r, a more detailed inves ti gation of the most hi ghly c ross-linked specime ns might be warranted in order to dete rmin e quantitatively the fate of these reaction products and their influence on the properties of the rubb er.
Th e sol co nte nt of th e cross-linked rubber consists of molec ules whi c h never become a part of the network. K. W. Scott [47] measured a sol conte nt of about 1 perce nt for nat ural rubb e r cross-linked with 1 phr of di c um yl pe roxid e a nd abo ut 0.1 percent when the a mount was 3 phr. Similar res ult s were re ported by Glaser and Eirich [33]. S till s mall e r amoun ts were found wh e n larger amounts of di cum yl peroxide were used. Con sequ e ntly the sol conte nt was assumed to be negli gi ble in all th e prese nt work.

.2 . Speci fi c Volume of Rubber
The s pecifi c volume of th e un vulcanized rubber (NBS Standard Refere nce Material 385b) at 298.15 K was 1.1074 c m 3 g -I. For si mplicity this valu e has bee n used for VI' throughout the prese nt s tudy. The actual specifi c volum e of the c ross-link ed rubbe r would be expected to be less than this value because of the dilue nt effe ct of the di cumyl pe roxide a nd th e c hange of volume on cross-linkin g. A dec rease of about 10 perce nt would be expected at the hi ghes t d egrees of c ross· linkin g. Th e s pecific volum e would also be ex pec ted to vary about 5 pe rce nt above and below th e valu e a t 298.15 K at the extre mes of te mp erature used in th e present s tudy.

Chain-End Correction
Th e co rre ction for danglin g c hain e nd s in effec tiv e for s upportin g a stress is di sc ussed in Section 3.1 of Part II [2] . Th e form of co rrection used there is exactl y th e sam e as that proposed by Flory [49]. H e used the quantity M(:-~" ell/ as a meas ure of cross-linking and so one finds by th e use of eq .   Later workers [50,51] have proposed sli ghtly different forms of the second te rm in brackets in eq (3.3.1) [52], while Mullins [20] gives experimental evidence that the factor should be It is likely that the factor in this form takes account of the effects of nonload·bearin g loop s, hitherto neglected , as well as chain e nd s. Use of th e rev ised factor would in crease the calculated value of effec tive mol ec ular weight by about 15 pe rcent. It would still be in the ran ge of reasonabl e values. Othe r re fin e me nts also give only s mall variation s in calc ulated molecular weight.
In figure 1 the whole c hain-end correction can be seen to result in s hiftin g the lin e to the right along the abscissa axis by only 0_14 phr, equivalent to a decrease of 9_36 /Lmol c m-3 on the ordinate axis _ This shift is so s mall that de termining the validity of the exact form of correction would require data of high precision exte nding over a wide ran ge of molec ular weights_ It would see m that more attention has bee n gi ven to this correction than would normally be justifi ed by its magnitude _ The dotted line in figure 1 is drawn neglec tin g th e correction comple tely, thus corres pondin g to an ass umption that the polymer was of infinite initial molec ular weight. The calculation for a finite molecular weight is of co urse gr eatly influ enced by th e correction for impurities, which in our work was more than twice that ascribed to chain ends.

Modulus at Low Degrees of Cross-Linking
In the experime ntal portion of the present s tudy (Part I [1]), the conclusions have been based on measure me nts where jp was greater than 0.45 phr, namely beyond the gel point. Whe n Il e is plotted against jp, as in fi gure 1, the consideration s already outlined yield the straight lin es s hown. The value of jp at which Il e is zero corres ponds to the amount of cross-linking agent whic h mus t be introduced to account for that required to establish a network . Only after thi s amount has bee n added can any additional cross-linking agent become effec tiv e.
At th e gel point th e number of moles of effective sub-chain s in zero (fi g. 1) and the modulus G is equal to G*, the e nergy component alone, so that G -G* = 0, as see n in figure 2.
In th e remainder of thi s section we s hall co nsider for the first time the actual observed behavior of the syste m at low degrees of cross-linking. Below th e gel point the network theory outlined pre viously c an make no predictions and we have been guided only by extrapolation . In this region experime ntal values are less reliable, since the creep rate beco mes high [53,54], and th e "equilibrium" modulu s is obtainable only by extrapolation to infinite time , for example by the method of Chasset and Thirion [23 , 55]. The modulus-temperature relation here is found experimentally to have a negative slope, as predicted by extrapolation from high degrees of cross-linking. This is evident in figures 2 and 7 of Part I.
Results reported by other observers [13, 15-20, 32-34, 36, 48, 56-65] agree with the present work in showing that the modulus increases linearly with crosslinking for the higher degrees of cross-linking. The line likewi se almost invariably has a slope greater than that predicted from the cross-linking. In the present work this slope was about 10.5 percent greater than predicted [2], while the literature values in the references just given range from 5 to 31 percent greater.
The line invariably has a positive intercept, ranging from 0.5 to 2 Mdyn cm -2. This is in accordance with The positive intercept and increased slope of the line are evide nt e ve n whe n the carbon-carbon crosslinks are form ed by expo sure to radiation [17 , 18, 66] rather than by free radicals res ultin g from the deco mposition of a peroxi de. Similar effects are evident in studies of silicone rubber cross-linked by radiation [57,58].
At th e lower degrees of cross-linking the actual measured values of modulus lie consiste ntly below the line, the de viation increasing as the c ross-linking is reduced. This behavior can be seen in the modulus of Chasset and Thirion's Specimen F , plotted in figure  8 of Part I. It is well verifi ed by other observers [13, 15, 17-20, 31, 32, 56, 57, 60, 61, 64]. The experimental values of modulus ofte n appear to decline to zero at a finite positive value of c ross-linking, and so metimes are even calculated as a negative modulus for the material to which no cross-links have been added [13 , 32].
An examination of the data suggests that as the cross-linkin g is reduced, th e value at whi ch the de viation from lin earity first occurs may be th e gel point. However, it is possible that the gel point is that at whic h the modulus actually becomes zero. Th e expe rime ntal and theoretical difficulties associated with consideration of the region near the gel point and below have already been mentioned.
It is interesting to compare our time-independent modulus value of Go = 1.87 Mdym cm -2 at 298.15 K, as meas ured on cross-link ed specim ens and extrapolated to zero cross-linking, with those derived from observations of a time -de pende nt modulus obtained from stress relaxation or shear creep expe riments on rubber actually containing no c ross -linking agent.
The latter values have us ually been obtained from a point of inflection in the plateau region of a plot of the logarithm of the ratio of stress to strain (or its reciprocal) against the logarithm of the time. Shear cree p s tudies on narrow-distribution synth etic cisisoprene polyme rs by Nemoto and co-workers [68] yielded s hear modulus values of about 0.8 Mdyn cm -2 with a plateau centered near -30°C (243 K). Stress relaxation studies by investigators in th e Polytechnic Institute of Milan on various polyisopre nes [69,70] s howed modulus values of 3 -4 Mdyn cm -2. Such nonzero values of modulus when no cross-links have been added have usually been ascribed to entanglements, which will be discussed in section 3.5.

. Entanglements
The linear relation between modulu s and crosslinking discussed in the preceding section has a positive intercept Go, the extrapolated modulus corresponding to no added cross-linkin g agent. Go has often been ascribed [19-21, 24, 31, 57,63,67,68] simply to entanglements functioning as effec ti ve cross-links, additional to but not much different in c haracter from those which have been introduced by the cross-linking agent. It would see m th at thi s s imple explanation mu st be abandon ed in vi ew of th e prese nt work , whi c h has now shown that th e modulu s has a ve ry appreciable energy component C* and that thi s is th e c hi ef factor determ inin g Co-The re lation be tw ee n th ese quantities is gIve n by eq (2 _5) of Part II Go=C* +SBT = A + BH + SBT = 2_67 -2.666 X 1O -3T. We are led to conclude that Go arises from interatomic for ces. Th e co nse ns us of opinion [71][72][73][74][75] has bee n th at I hese forces are largely intrac hain (" intramolec ular" ) in o ri gin.
The data indi cate that th ese forces (whatever th eir ori gin ) in c rease sli g htl y with in c reas in g c ro sslinkin g, as might be expected. If e ntangle me nts serv e to increase th ese forces, they may well hav e an important part in givin g ri se to Ih e inte rce pt.
On the ot he r hand if e ntan gle me nts act merely as pse ud o effec tive c ross-link s, th ey ca n contribute onl y to th e e ntropy compon e nt of the modulu s. In expe rime ntal term s, th ey can in c rease the slopes of th e G, T lin es of figure 1 of Part II but wo uld not affec t th e inte rce pts of th ese lin es.
Previou s observation s of the relation betwee n modulus and c ross-linkin g, with a few e xce ption s [29,76,77] have bee n Limited to measure me nts at a sin gle te mperature or ove r a s mall range . Consequ e ntly th ey have not furni s hed informati o n regarding th e relative contribution s of e nergy and entropy_ Th e theory of e nta ngle me nts has bee n e xten sively di sc us sed in rece nt years [78][79][80][81][82][83][84][85][86][87]. Th e co nsid erations have now progresse d con s ide rabl y beyo nd the s impl e id ea of ps e udo c ro ss -link s jus t mentioned . For example topological e ntan gleme nts have bee n distinguished from couplin g loci arising from inte ratomic forces, and som e e ntan gle me nts are regarded as trapped betwee n c ross -Ijnk s while others are untrapped. Furthermore e ntangle me nts are important in determining the vi scoelasti c and othe r properties of a given polym er. We res tri c t ouselves here to considering their effec t on th e modulu s. Some th eo re ti ca l treatme nts predi ct th at the number of e ntan gle ments s hould in c rease lin early with an in crease in th e numb e r of c ross-link s, while others co nclud e that it s hould be inde pende nt of th e crosslinkin g. The prese nt work would favor th e form e r co nclusion, in vi e w of the fact that th e obse rved value of th e co nstant 5 in eq (1.1) is greater than that comp uted from th e molec ul a r con s tants in Part II. If thi s in crease is ind ee d du e to e ntagle me nts and not to so me defec t in th e ass umpti ons, one would co nclude Ih a t th e number of e ntan gle me nts is co ns is te ntl y about 6.7 percent of th e numbe r of cross -link s. We know of no independe nt me th od by whi c h to verify thi s conclusion.
If e ntangl e me nt s are regarded as te mporary crosslink s [30,67,88] which are effec tive only for a limited tim e afte r th e appli ca ti on of a stress, th ey need be co nsid e red o nl y whe n th e re is appreciable c ree p. Th e signifi ca nt portion s of th e prese nt work we re don e und e r co nditi o ns wh ere th e c ree p was negli gible . Co nsequ e ntly , he re we ca n neglec t th e effec t of s uc h te mporary c ross-link s .
The sco pe of the prese nt pape r does not permil furth er di sc ussio n of e ntan gle me nts, othe r th a n to re peat that th e y are not functioning as pseudo crosslink s ma kin g a major contribution to th e value of Co.

3.6_ "Front Factor" and Efficiency
On e of th e most significant conclusions of the prese nt work is th e prediction of th e value of th e co ns tant 5 in eq (1.1) as equal to 2 R / (lOOM ,tijr)' by eq 4.3 of Part II.
R egardless of wh e ther th e 6.7 pe rce nt excess of th e meas ured ove r Ih e calculated valu e is ascribed to defective ass umpti ons o r to th e prese nce of e nt a ngleme nts, as ha bee n don e in a pre viou s section, it is c lea r th at th e e xte nt of th e num e ric al agree me nt appare ntl y e limin ates th e necess ity for se veral re fin eme nts a nd correc tion s whi c h might be mad e in th e s impl e th eo ry outlin ed he re.
Th e re has bee n co ns id erable di sc uss io n regardin g a Most rece nt a uthors ha ve includ ed a diffe re nt " front fa c tor" (rf)/(r~) where (rf) is th e mea n s quare valu e of th e di s place me nt lengt h of th e s ub-c hain in th e isotropic un strained state and (r~) is th e mean-s quare value of the displacement le ngth in th e unperturbed state [74]. The latter quantity vari es with te mperature.
The assumption that each molecule of d ecomposed di c umyl peroxide gives rise to one cross -link in natural rubber is ge nerally accepted [21, 31, 94-97] but th e efficie nc y E has occasionally been stated to be less than unity under certain circ um stan ces [27 , 28, 60], (especially if no account is take n of the amount wasted by reaction with impurities). Scission during c ure would also redu ce the efficie ncy, but has rece ntly bee n s hown to be negli gible [28].
Th e introduc tion of a " front-factor, " </> whatever its origin, and a n e ffic ie nc y fac tor E into eq (3.9) of Part II or eq (2.3.2) of th e present pape r would res ult in th e s ub stitution of $ER for R, th e gas co ns tant , in all th e relations co ntainin g R. Th e prese nt res ults indicate that the produ c t </>E, e ve n without th e e ntan gle me nt correc tion , co uld not diffe r from unity by more than a bout 7 pe rce nt. A signifi cant variation of thi s quantity with temperat ure would have bee n obser vable as a deviation from linearity in the graph of aG/afp agains t T and of aG/aT against fp (fi gs. 6 and 7 res pectively of Part I).
If the front fac tor cf> is taken as 0.5 the effi ciency would have to be an unlikely 200 perce nt (i.e., each molecule of decomposed peroxide wo uld have to furni sh two cross -li nks yielding fo ur additional s ub-chain s)_ Such a counter balance of fac tors does not see m probable and we prefer to consider that the prese nt results indicate values of unity fo r both cf> and E under our conditions of c ure.

Resolution of Components of Modulus
Many of the concepts developed here can be more readily unders tood when t hey are utilized to resolve the modulus at fp = 1 phr, and T = 298. 15   amount of dicumyl peroxide wasted by reaction with impurities is shown as a horizontal displace me nt of 0.31 phr, while the effect of dan gling loose en ds of the rubber molecules is given by an additional horizontal displacemen~ of 0.14 phr to the gel point. The second component, also an energy term independent of temperature, corresponds to the term con taining the constant H and is the smallest of the four componen ts.
The third term, an entropy term proportional to the temperature, is the one which can be calc ulated as RT times the density of effective s ub-chains attributable to the added dicumyl peroxide. Finally, there is a fourth term, which is also an entropy term proportional to the temperature and the cross-linking. This term can probably be ascribed to the effect of en ta ngle me nts, functioning as pseudo cross-links_ The lines shown in figure 3 are to be regarded as schematic, since they are the extrapolations of lines dete rmin ed largely at hi gher valu es of c ross·linkin g. In the present work (exce pt in sec. 3.4), little significance has bee n attached to actu al ex perime ntal valu es in the region of the gel point a nd below.
In view of the fac t that a large portion of th e modulu s ari ses from an energy componen t G* (es pecially at low degr ees of cross-linkin g), it is n ot s urpri s ing th at th e fo rm of stress -s train rela tion derived from entro p y con siderati on s alone giv es only an a pprox im ation to the experime ntal da ta [63,73,90]' th e value predi c ted bein g cons iste ntly hi ghe r th an the obse rved stress [98 , 99] . It will be note d that th e prese nt work has require d no ass umptions whatever a bout the equa tion of state or form of the stress -strain relation outside the range of infinitesimal deformatio ns.

s. Conclusions
The qualitative and qu antita tive agree me nt of predic tion s and res ults de monstrated in the pre viou s sections is a strong confirm a tion of th e esse ntial validity of a ll th e extre mely simplifi e d molec ular consid eration s involved, in clud ing the general as pects of th e stati stical theory of rubber elasticity. We know of no pre vious experime ntal study exte nding ove r as wide ranges of cross-linking a nd te mpe rature . In fact the c ross-linking and te mpe rature have been varied simult an eously on only a fe w occasions in pre vious work .
An important advantage of the prese nt work over many pre vious studies is the fac t that measure me nts are made a t very small deformations. Thus the res ults are e xpresse d in te rm s of th e modulus, de fin ed as the limiting value of the ratio of stress to strain at ze ro defo rmation. Conse que ntly, the res ults are indepe nde nt of the stress-strain relation or equation of s tate_ This means that no conside ration needs to be given here, fo r exa mple to the differe nce between the stress -strain relation predi cted by the st atis tical theory of rubbe r elasticity and that given by the M ooney-Rivlin e quation or the e mpirical equa tion of Martin , Roth , and Stie hle r [99].
The present study has shown that the modulus G includes a considerable component arising from inte rnal e nergy c hanges as well as that arisin g from entrop y c hanges. The e nergy component at room t emperature is of the order of half the total whe n the degree of cross· linking is that normally used with dic umyl peroxide r ubb ers.
It is concluded that the nonzero value of the modulus when extrapolated to zero cross-linkin g is due to the e nergy com pone nt of the modulus r athe r than to e ntangle me nts. E ntanglemen ts actin g as pse udo-cross-link s wo uld ser ve to in crease only the e ntropy compone nt.
T he gel point, defined as the minimum degree of cross-linkin g required to for m a network , may be loc ated experime ntally as the cross-lin king at whic h the slope of the modulus-te mperature relation is zero. The value of the modulus G at the gel point is not zero, but is the e nergy component under thi s condition; the e ntropy compon ent of G at the gel poin t is zero.
The amount of dicumyl pe roxide required to crosslink rubber to the gel point is the s um of that wasted by reaction with impurities in the rubber and that required to give one cross-link for each rubb er molecule. Th e former quantity was about twice the latte r in the work reported here.
The entropy component of th e modulu s as determined from reported values of equilibrium swellin g by the Flory-Rehner equation, is found to be signifi cantly larger than that determin ed from mec hani cal meas urements. However, the quantity co mputed is smaller than the sum of the entropy and e ne rgy compon e nts as de termined from cross-linking cons ideration s or from mechanical measure me nts. It in c reases linearly with increase of cross-linking a t a slightly greater rate than the modulus or the e ntropy compone nt of the modulus.
It is concluded that the " front factor" some times introduced in stati s tical th eory conside rations c annot differ from unity by more than about 7 pe rce nt. Th e differe nce is eve n less than thi s if allowan ce is mad e for e ntan gle me nts functioning as pseudo-c ross-links.

Numerical Values of Constants
For conven ien t refere nce a ll th e num eri ca l valu es use d in ca l· c ul ati ons in th e prese nt p ape r a re give n he re.