DYNAWC MECHANICAL PROPERTIES OF CROSS-LINKED RUBBERS 353
Dynamic Mechanical Properties of Cross-Linked Rubbers. VI. Poly ( dimethylsiloxane) Networks Neal R. Langley' and John D. Ferry
Department of Chemistry, Unicersity of Wisconsin, Madison, Wisconsin 53706. Receiced March 12, 1968
ABSTRACT: The viscoelastic properties of three cross-linked samples of poly(dimethylsiloxane), prepared by high-energy electron irradiation of an initial polymer with nearly uniform molecular weight of 5 X I@, have been studied by dynamic and creep measurements. The sample with least cross-linking was also studied after extraction of its sol fraction (56.5z).The frequency and temperature ranges were 0.05-600 cps and -42 to 102'; the creep experiments extended to durations of 5 days. The creep data were converted to the corresponding dynamic viscoelastic functions at very low frequencies. All data were combined by reduction to 25" with shift factors calculated froin earlier viscosity measurements by Plazek on an uncross-linked sa nple of similar molexlar weight. Only the region of secondary (low frequency) loss was encompassed, the primary transition zone lying at higher reduced frequencies. The equilibrium compliances, J,, were obtained by fitting the creep data to a modification of the empirical equation of Thirion and Chasset. The compliance corresponding to the entanglement network was obtained by diflerence between the equilibrium compliance and the integral over the retardation spectrum in the secondary loss region: log J,\. = -6.47, corresponding to an entanglement spacing of 220 chain atoms. With increasing cross-linking. the secondary (low frequency) maximum in the loss tangent decreased in magnitude and shifted to higher frequencies, as observed for several other rubbers. From swelling and sol extraction measurements. and the assumption that both cross-linking and chain scission by high-energy electrons were proportional to the radiation dose. the parameters needed to compare the ratio J,.v/J, with a recent theory of Langley were obtained. The agreement was satisfactory. indicating that the low-frequency loss mechanism might be attributed to the relaxation of untrapped entanglements on floating structures loosely attached to the three-dimensional network. An approximate extension of this theory is qualitatively successful in explaining the frequency dependence of the storage compliance at intermediate frequencies.
In
previous paplers of this s e r i e ~ , we ~-~ described viscoelastic properties of natural rubber, 1,4-polybutadiene, and a styrene-butadiene random copolymer rubber, all cross-linked to varying extents; the investigations encompassed both the transition zone and the region of low frequencies (or long times) where anomalous relaxation mechanisms are evident. In the present study, similar data are reported for three poly(dimethy1siloxane) networks, cross-linked by high-energy electrons, a n d one such network from which the sol fraction had been extracted. Because of the high segmental mobility of this polymer, the transition zone was not covered in the range of frequencies and temperatures experimentally accessible. However, the low-frequency losses were prominent. They can be quantitatively interpreted in tenms of the relaxation of entanglements o n branched structures loosely attached t o the network, as calculated from relations derived in a n accompanying communication. Experimental Section The samples were generously provided by Drs. A. M. Bueche and E. E. Bostick of the General Ekectric Company. The parent polymer had been decatalyzed after polymerization and thoroughly dried under vacuum; its molecular weight as determined in their laboratories by osmotic pressure in benzene ssolution was approximately 500,000. by light scattering in benzene 508,000, and by intrinsic viscosity ( I ) Weyrrhaeuser Foundation Fellow, 196551967, To whom inquiries should be addressed a t Battelle Northh est, Richland, Wash. 99352. (2) (a) J. D. Ferrq, R. G. Mancke, E. Maekawa, Y . &magi, a n d R . A. Dickie, J . Phj.s. Chem., 68, 3414 (1964); (b) E. Maekawa, R . G. Mancke, and J. D. Ferry, ibid., 69, 2811 (1965). (3) R. A . Dickie and J. D. Ferry, ibid., 70, 2594 (1966). (4) R . G. Mancke and J. D. Ferry, Tram. Soc. Rheol., i n press.
in benzene 530,000; these values imply a quite sharp molecular weight distribution, which is important in comparing the viscoelastic properties with theoretical calculations.: This polymer, in the form of sheets about 1 mm thick with outer wrappers of Ethocel, was exposed to high-energy electrons; samples A-1, A-2. and A-4 correspond to doses of 1. 2, and 4 M R (megaroentgens). From the experiments of Bueche and collaborators.6 1 M R is expected to produce about 2.4 X 10-6 mol of cross-links/g, but from swelling and sol fraction determinations a somewhat smaller degree of cross-linking was deduced, as described below. For swelling and sol extraction measurements, samples (from 0.16 to 1.3 g) were immersed in 25-50 cc of reagent grade toluene, supported by Teflon bases in Petri dishes. The solvent was replaced every few days and the cumulative extract was evaporated to dryness and weighed. The degree of swelling was determined by weighing the drained swollen samples after 8 and after 14 days, with agreement within 1 %. The extractions were continued for 37 days (15 extractions) for samples A-1 and A-2 with no additional increments to the sol fraction, and for 78 days (19 extractions) for sample A-4, with small additional increments to the sol fraction. A control series of evaporations of the solvent gave a small correction for nonvolatile residue, though the amount was within the 0.001 limit specified by the manufacturer. The gels of A-2 and A-4 were subsequently deswollen, dried, and weighed. and the sum of sol and gel fractions agreed very well (within 0.3%) wi:h the original sample weights. An extracted gel of A-1, which had a gel fraction of only 0.44. was very carefully deswollen on a mercury surface to provide a sample with suitable shape for measurements of viscoelastic properties (sample A-1-E). to compare with the original unextracted sample. In calculating the polymer volume fraction r? of the swollen gels, the volumes of polymer (5) N. R . Langley, Mucromolecules, 1, 348 (1968). (6) L. E. St. Pierre, H. A . Dewhurst, and A. M. Bueche, J . Polym. Sci., 36, 105 (1959).
354 NEALR . LANGLEY A N D JOHND. FERRY
and toluene (density 0.861 g!ml at 25l) were assumed to be additive. Disk-shaped samples, 1.75 cm in diameter, were cut for viscoelastic measurements. Their densities, measured pycnometrically, were 0.972 i 0.001. The Fitzgerald transducer' was used for measurements of the storage ( J ' ) and loss (J") components of the dynamic shear compliance in the range from 24 to 600 cps, over the temperature range from -42 to 102 . At lower temperatures, crystallization occurred. The Plazek torsion pendulum8 was used for dynamic measurements in the range from 0.05 to 6 cps and also for creep measurements over time periods up to 5 X lo5 sec. In addition to improvements in the torsion pendulum previously reported,3s9a fixed secondary mirror was installed to intercept a lower portion of the light beam incident on the primary torsion element mirror; this reflected a stationary image near the image of the primary mirror, within the traverse of the pair of solar cells. The change in position of the primary moving image was always referred to that of the stationary image rather taken directly from the scale of the lathe tool mount.g In this way, small irregularities during long creep runs. attributable to changes in solar cell efficiency, were largely eliminated. Also, to obtain creep data in a time range including shorter times than were previously possible (6 sec to several minutes), one edge of the beam reflected from the primary mirror was directed to a single solar cell whose output was amplified and continuously recorded on a Sahborn Type 151 recorder. The amplitude of the resulting trace was calibrated against a standard creep detefmination. Further details of procedure are given elsewhere."' Results Both creep and dynamic data were reduced to a temperature of 25" by shift factors uT based on viscosity data of Plazek" on uncross-linked poly(dimethy1siloxanes). Above -21", these correspond to a n Arrhenius temperature dependence with a n activation energy of 3.65 kcal; below -21 values were interpolated from Plazek's measurements. The thermal expansion coefficient was takenI2 as 8.6 X 1 0 P deg-'. For the creep measurements, it was necessary to make small arbitrary corrections to the magnitude of the creep compliance at each temperature to achieve superposition; the deviations were attributed to errors in determining the sample thickness, which was unusually small for this type of measurement. These corrections were chosen t o make the individual reduced creep curves agree in magnitude with the torsion pendulum dynamic data (all of which superposed without any correction) after conversion of creep to storage compliance as described below. Otherwise, superposition of viscoelastic data at various temperatures was in every case very satisfactory. Creep and Equilibrium Compliance. In Figure 1, the reduced creep compliance J17(r)is plotted logarithmically against reduced time for all four samples. O,
(7) E. R. Fitzperald, Ph,rs. RcP.,108, 690 ( l Y 5 7 ) . (8) D. J. Plazek, M. N. Vrancken, and J. W. Berge, Trans. Sac. Rheol., 2, 39 (1958). (9) I
x x
0.5636 2.225 0.0020
I . 127 2.254 4.450 8.90 T, 0,0665 0.199 v (from wr) X lo6 0.30 9.2 28 ve.v x 106 118 119 123 YIVe.V 0,0025 0.077 0.23 a Assumed equal to value for sample A-4 (cf: ye,,- values).
p q
104 104
that ye.\- is the density of effective network strands terminated either by chemical cross-links or by entanglements, either trapped or untrapped, whereas v is the density of effective network strands terminated by chemical cross-links or rrupped entanglements. The values of J , and Jes and their ratios are given in Table I. The calculations of v and were made by the following procedures. (a) At equilibrium swelling, the polymer volume fraction in the swollen gels, c?, was used to calculate v, the moles of effective network strands per cubic centimeter of total original polymer (sol plus gel fractions) by the equation
v
=
-[In (1
-
c2)
+ cp + x1c:!2]Mig/VL(MIg2'3 c 2 1 ' 3
- cn/2) (3)
where w, is the gel fraction, determined from sol extraction measurements, V , is the molar volume of solvent, and x1is the thermodynamic interaction parameter. Equation 3 is the Flory-Rehner equation18 which has been generalized here to apply to lightly cross-linked networks with significant sol fractions. The w, factor in the numerator is introduced to yield v based o n the total polymer volume rather than the gel volume; the w g 2 factor is used because the sol fraction serves to dilute the gel during cross-linking, causing greater ~ w e l l i n g 'than ~ if the gel were cross-linked while undiluted. Values of xl were estimated from a relation, implicit in Figure 3 and eq 1 and 3 of ref 6. X I
=
0.465
+ 0.0160~RT
(4)
where ue\ is the effective network strand density corresponding t o the plateau value J,, and v is the strand density at mechanical equilibrium. We seek to show
For each sample, values of x1 and u were calculated which satisfy both eq 3 and 4. The values of L ' ~ , w,, x,,and u are given in Table 1. (b) Values of u were also calculated from J, by the elementary equation of rubberlike elasticity, v = I,'J,RT, and are included in Table I. They agree
(16) W. J. Bobear, J . Polj,ni. Sci., Part A-2, 4, 299 (1966). (17) G . C. Berry and T. G. Fox, Adcuri. Poljrn. Sci., 5, 261 (1967).
( 1 8 ) P. J. Flory a n d J. Rrhner, Jr., J . Chen?. Phj,s., 11, 521 (1943). (19) P. J. Flory, ibid., 18, 108 (1950).
DYNAMIC MECHANICAL PROPERTIES OF CROSS-LINKED RUBBERS357
Vol. I , N o . 4, J L ~ I J . - A U ~ U I968 U
moderately well (within 20%) with those derived from eq 3. (c) Values of v were also calculated from eq 19 of ref 5 . v
=
+
(C~O,!MO)M',T, 2eT, 'I
(5)
In this treatment, 11 and q are the probabilities that a randomly chosen monomer unit of the original polymer has undergone scission or is chemically cross-linked, respectively, 2e is the maximum potential contribution of entanglements to v, and T, is the probability that a randomly chosen entanglement locus is trapped because all four strands radiating from it lead to gel. The probability T,. for the uniform initial molecular weight M is given by eq 17 of ref 5 as
T,
x l [ l - 2(1 - p-!'f'),'yP
+
+ e-U'']Q
(6)
+
Here x = C ~ - J ( / , ) q w g ) , J' = / I qwg, and P = MjMo. To evaluate thle parameters p and q, a n additional relation was employed, eq 6 of ref 5 for a uniform initial molecular weight. IC,
=
2x[1 - (1 - e-"")/s.Pl x y [ l - 2(1 - e-"")/.vP
+ e-""]
(7)
It was assumed that both p and q were proportional to the radiation dose for the small doses used here, and the experimental values of ivK for samples A-1 and A-2 were used to evaluate /I and y by the trial and error scheme of ref 5 . Again assuming proportionality to dose. the value of w, calculated from eq 7 for sample A-4 is 0.903; this is in good agreement with the experimental value of 0.925 especially since the latter may be slightly too high as traces of sol were still being extracted at the end of the extraction experiment with this sample. The resulting values of 11, y, and T, are included in Table I. The empirical parameter E was chosen to make v from eq 5 give the best agreement with that from swelling, eq 3. Actually, the value thus selected, 6.0 X 1 O-" mol'cc, corresponds to a n entanglement spacing of 220 chain atoms, in excellent agreement with the estimate from the magnitude of Je.,-. Values of v calculated in this manner are included in Table I . Finally, ue,- was calculated by considering the entanglement network to be formed by the total entanglement contribution, 2 ~ as , well as the chemical cross-links. In eq 19 of ref 5 , the first term o n the right side represents the strand density of the crosslink network. Thus, the strand density of the entanglement network is given by the same term where functions of the variable q are replaced by the same functions of the variable ( q 2eM0.'p).
+
Eq 6 and 7 were used to evaluate these functions and the vv\ values calculated for the three samples are given in Table I . The ratio ujve.\. from these statistical calculations, the last row in Table I, agrees reasonably in magnitude with J,\,'J,, supporting the view that the secondary loss mechanism is indeed associated with the relaxation of entanglements with at least one of the four radiating strands leading to a finite-sized structure. Some additional coinnients may be made on these
results. The values of y imply about 38% less crosslinking efficiency of the high-energy electrons than expected from earlier work.6 However, this is perhaps not too serious a discrepancy. It is of interest that v? \- remains almost constant with increasing radiation dose, justifying the assumption that JeV can be taken to be the same for all samples. If it were not for the simultaneous scission, v , . ~ would increase somewhat more with the formation of additional chemical crosslinks. Values of V / V , . ~ calculated for the above cornparison are sensitive to the values of T, since the 2€Te term in eq 5 is always at least 0 . 8 ~ . Thus, values of T , calculated statistically by eq 6 are confirmed by the agreement of uiues with Jes)Je and of u from swelling results with v from eq 5 over the hundredfold change of u. The effect of a constant front factor in the theory of rubberlike elasticity (g in eq 20 of ref 5) is a variation, inversely proportional to the front factor, in v calculated from J , and (because of the origin of the xL values) in v calculated from swelling results. The values of e and ueT would also vary in nearly the same proportion; thus, agreement of the quantities compared above is not affected by the value of the front factor. It is of particular interest that the proportion of trapped entanglements, T,, is exceedingly small for the very lightly cross-linked network and is only 0.2 for the most highly cross-linked network studied here. Qualitatively, this feature is important in interpreting the general origin of low-frequency losses observed in various rubbers. 2 - 4 Unfortunately, quantitative calculations cannot be applied to the latter because of insufficient information. Frequency Dependence in the Region Attributed to Untrapped Entanglements. Although the magnitude of Je/Je.vis predicted rather satisfactorily by the theory embodied in eq 5-8, the location of this secondary dispersion o n the frequency scale and its detailed frequency dependence are much more difficult to treat theoretically. They will depend o n the size distribution of floating structures bearing untrapped entanglements, insofar as these features play a role in the coupling with the permanent network. The probable importance of branching in such structures can be inferred from observations that the longest relaxation times (and viscosities) are enormously increased by moderate branching in systems where the relaxation mechanisms are dominated by entanglement coupling. 4 * 2 0 In any case. the processes responsible for relaxation times extending out to times of more than 10: sec, as evident in Figures 1-3, must be far slower than those in unvulcanized polymer of similar molecular weight; the data of Plazek (cJ Figure 2) indicate that the latter d o not extend to times higher than l o 2 sec. Thus if linecir dangling free ends contribute to the slow relaxation mechanisms, their entanglements with the network structure must reduce their mobilities to a far greater degree than d o similar entanglements in a n uncrosslinked system. An approximate calculation"' based on the shortest chain lengths extending from untrapped entanglements, (20) G . Icraus and J. T. Gruvcr, J . P o ( . t n . Sci., Purl A , 3, 105 (1965).
358 D. L. CHRISTMAN AND G. I. KEIM
without consideration of the degree of branching, reproduces qualitatively the frequency dependence of J’ observed in Figure 2 at higher frequencies. Here it is assumed that an entanglement from which the shortest radiating strand is 112 monomer units long will contribute to the effective network only at frequencies exceeding a critical value w(m). The function w ( m ) can be determined from the frequency dependence of J’ for one sample (A-2) and the other J’ curves can be predicted from it, but the calculation is restricted to m
