Thermodynamics of the Densification Process for Polymer Glasses

A quantitative description is given for the densification process of glasses resulting from glass formation at elevated pressures. Phenomenologieal relations are derived, or justified, which allow estimation of the densification rate κ′ (with respect to formation pressure) from various thermodynamic quantities and glass transition behavior. In addition, the estimation of K′ may be facilitated by the application of the hole theory of Simha and Somcynsky. Using these relations κ′ is estimated, and the results from the different methods are compared for data from 23 different organic polymers with glass transition temperatures ranging from 150 to 455 K. The amount of densification appears to be limited by the apparent convergence of the glass temperature and effective decomposition temperature with increasing pressure. Some estimates of limiting values are presented. Finally, changes of refractive index resulting from densification are estimated from the observed, or predicted, densification rates.


Introduction
The density of a glass, as well as ce rta in other properties, depe nd upon th e thermodynamic histoll' by whi ch the glass is form ed. For exampl e, as s hown schema ti cally in fi gure la, an amorphous polymer subj ected to an elevated press ure in the melt, followed by isobaric cool ing a t constant rate to a temperature well be low t he glass te mperature, T g, and then depressurized, will have a large r dens ity than that obtained by isobaric cooling at the sa me ra te at atmospheric pressu re to the same temperature in the glass .From the former procedure the pressure induced densification rate is defined as K' = -(l /V)(aV /a p'lr,p,k (1) where V is the volume at tempe rature T and pressure P, and P' is the form ation pressure maintained during constant rate of cooling k.Note that this definition parall els the usual one for the isothermal compressibility, K = -(l/V)(aV /a ph,p' ,k, (2) the difference being that P and P' are interc hanged.
It is expected that th e fina l de pressurized vol ume in the glass will lie between the atmos pheri c and th e pressurized value , as shown in figure lao It is the n clear that the inequality, where Kg IS the compressibility of the glass, IS obeyed.Although we may intuitively expect this relation to hold, as apparent from experiment, we do not know of any proof.
A well known alternative method of densifying glasses is simply to decrease the cooling rate as illustrated in figure lb.In principle one can obtain the same volume in the glass by this procedure as by elevating the pressure, except that the times required for the former are much longer.For example, it is estimated [1]1 that a poly (vinyl acetate) glass obtained by isobaric cooling at 800 bar2 in 8 hours would require 500 years to reach the same volume at the same terminal temperature by cooling slowly at atmospheric pressure.It should be recognized, however, that the states of glasses at the same volume, temperature, and pressure, but obtained through different histories, are not necessarily the same.As pointed out by Bree and coworkers [2], volume changes during isobaric-isothermal volume relaxation [3] have a large effect on relaxation times for creep compliance, whereas almost no effect is observed from volume changes obtained by isobaric cooling at elevated pressures.Accordingly, it appears that the state of a glass is not determined by its volume , temperature, and pressure alone.Moreover, pressure induced densification does have an influence on physical properties.According to the data of Dale and Rogers [4] over a 5 kbar range, the compressive modulus of polystyrene appears to increase slightly with formation (or molding) pressure, leveling off at higher pressures, with the yield stress going through a maximum between 1 and 1.5 kbar.Wetton and Moneypenny [5] have studied the dynamic mechanical and dielectric properties of several polymeric glasses formed at pressures up to slightly beyond 5 kbar.Both the real part of Young's modulus and its loss tange nt, as well as the real part of the dielectric constant, increase with formation pressure.For poly(vinyl acetate) McKinney and Goldstein [1] have observed a 3 percent increase in the bulk modulus at 0 0 C, corresponding to a formation pressure of 800 bar.This difference increases with decreasing temperature.
Thermal prope rties also seem to depend on the amount of pressure induced densificatioll.Although the heat capacity C p is found to be independent [6] of formation pressure, the enthalpy H seems to vary significantly at formation pressures above a certain value.According to Price [7], very little change in the enthalpy of poly(methyl methacrylate) is observed up to about 800 bar, followed by a nearly constant rate of increase of about 0.015 cm 3 jg up to 3 kbar, their maximum value.For polystyrene [8], the data have been evaluated as MI = H(densified) -H(normal) first decreasing slightly and then going back to zero at about 800 bar, followed by an increase with nearly constant slope up to the maximum pressure.Weitz and Wunderlich [9] have also observed this behavior and interpreted it in terms of two opposing mechanisms arising from holes and rotational isomers.It is not clear, however, that the apparent negative values of MI obtained by experiment are significant.
The purpose of this paper is to describe the thermodynamics of the pressure induced densification process by applying both phenomenological and molecular theory.Simple phenomenological relationships are deri ved between the densification rate K' and other thermodynamic propelties for which values are more readily available in the literature.Moreover, it is shown how the hol e theory of Simha and Somcynsky [10] may be used to facilitate the estimation of the densification rates for polymers using a minimum amount of experimental information.In both cases the derived relationships are tested using appropriate experimental data.An example of the utility of these results is demonstrated by estimating the change in the index of refraction corresponding to changes in molding pressure, assuming that the index of refraction is related to the volume by the Lorentz-Lorenz equation.The results have potential application to the adjustment of the refractive indices of lenses by varying the moldmg pressure.

Phenomenological Relationships
Two types of thermodynamic histories, shown schematically in figure 2, are pertinent to the development of the phenomenological relationships for pressure induced densification.In the first (Fig. 2a) the PVT surface of the glass is obtained from repeated isobaric cooling runs at the same constant rate, but at difference pressures, with all pressure changes occurring in the melt prior to each run.This procedure is called the variable formation history because the structure of the glass is different for each experimental pressure (which is the formation pressure, since P = P').The glass transition at each pressure is assumed to occur at a constant mean relaxation time.Hence, the intersection of the liquid and glass PVT surfaces gives the proper T g(p), from which dT gjdP is expected to approximate that obtained from the dynamic mechanical and dielectric frequency-temperature-pressure superposition.On the other hand, as a consequence of the varied structure, the glass PVT surface is not proper in the thermodynamic sense.With the other history (fi g. 2b) the glass is form ed also by isobaric cooling at consta nt rate at an a rbitra ry pressure, p' (which is usua]] y atmospheric , but ele va ted in fi gure 2b to illustrate the more ge neral case ).At temperatures well below T g , where viscoelas ti c relaxation times are large in comparison to effecti ve expe rimental times, a thermod ynamicall y reversibl e PVT surface for the glass is obtained by observing the volumetri c response to "fast" changes in te mperature and pressure .Since all of the data in the glass pe I1ain to the same P' , the PVT surface gives the proper values of the derivable the rmodynami c quantities (for example, thermal ex pansion , isothermal compressibility, and internal pressure).The intersection of the liquid and glass surfaces defin es the fi ctive temperature Tt(P, PI).
The princip~l distinction in procedure be tween the two histories is that with varia bl e formation all pressure c ha nges are made in th e melt, wh ereas with constant form ation they are made in the glass.Note that th e numbe r of inde pendent variabl es is differe nt for Tg(P) a nd Tt (P , PI ).The redundancy of using two arguments in th e form e r a ri ses from th e fa c t tha t the form ati on a nd ex pe rime ntal pressure are always ide ntical.Accordingly, T g may be regard ed as a s pec ial case of Tt whe n P = P' .The implic it argument k is de leted here becluse onl y one value appplies to these di scuss ions for eac h case.For furthe r deta il s and interpretations of these hi stories, see Ref. [1].
In all of th e sc he mati c diagrams in this pa pe r the glass transition is sh own as a discrete inte rsec ti on.With iso bari c cooling at cons tant rate through Tg a gradual tra nsition process is observed .Th e di sc rete intersecti ons s hown correspond to those obtaine d by extra polati on of th e equilibrium isobars a nd the isoc hro nal (nonrelaxing) ones for the glass.
We now proceed to evalua te the th e rmod yna mi c di agra m in fi gure 3, in ord er to determine relationships fo r K' in te rms of other measured qua ntities .Volume A is obta ined by isobari c cooling at co nsta nt rate and atm ospheri c pressure (P = P' = 0).Volume B is reac hed by pressurizing to P = P' = M in the melt, foll owed by iso baric cooling at the same constant rate as for A, with s ubseq ue nt depress urizing in the glass at the same terminal temperature as for A. Note that Tg (in li eu  of Tl) applies here, since the trans ition is observ ed at the formation pressure (P = PI) in both cases.The isobaric extension of VB with increasing T (see das hed line ) to its intersection with the liquid line yields the fic ti ve te mperature

Tt(p, PI) = Tt(O, boP).
gIn summing gthe thermodynamic contributi ons for small changes in T and P near T J,f) in the ra nge where linear approximations are valid , we find where a is the usual isobaric thermal ex pa ns ivit y, the Ll's ind icate differences as shown on fi gure 3, and the subscripts L a nd g pertain to liquid a nd glass.For sma ll c hanges eq (1) may be writte n in th e form (4) whe re the Ll's he re indi cate th e usua l diffe re nces 111 the respec tiv e q uantiti es between liquid and glass.Since a long Tt, [11], K' may be expressed in te rm s of th e d iffe re nce befw een the two tra nsition rates, i.e. ,

(5b)
Eq uati on (Sa) may a lso be written as a n E hre nfest-type relati on, viz., (5c) which is consiste nt with the experime nta ll y observed inequality provided the de ns ifi cation rate is non-negative.Expressions full y equivale nt to eq s (5) have been derived by Goldste in [11 ] a nd give n previously in Ref. [12].From eqs (5) it is evident (as also pointed out by Goldstein [11]) tha t th e necessary and sufficient condition (assuming Lla =t-0) for th e PVT surface to be independent of formati on pressure is the validity of the first Ehrenfest equation The analogous argument applies to the entropy sUl{aces.Since the second Ehre nfest equati on , (7) where C p is the usual heat capacit y a t co ns tan t press ure, appears to be a good a pproximati on [11 , 13], there should be a si ngle entropy sUI{ace with respect to formation pressure in contras t to the manifold/surface observed for volume.This view is confirmed by the DSC 3 measurements of Y oUltee and Cooper [6] on normal and d ensified pol ystyrene, which reveal no significant effect on the thermal propel,ties of glasses by vitrification at elevated pressures.The authors did find some differences in the thermal behavior between these properties and those from vitrification by isothermal compression ; however, these were attributed to inhomogeneous freezing processes during compression.Accordingly, if eq (7) is a good approximation, it leads to a conve ni ent experime ntal de termination of the initial (P = 0) value of dT JdP through volume-temperature and heat capacity measurements required at atmospheric pressure onl y.Equation (7) will be tested by means of experimental data later in this paper.
As stated above K' may be determined (near T g) from the values of Lla, dT gjdP, and LlK using eq (Sa).The relative difficulty in obtaining these quantities experimentally increases in the order give n above, as does the difficulty of obtaining their values from the literature.For these reasons it is desirabl e to be able to estimate LlK (or LlK/Lla) independently of existing PVT data.It will be shown how the hole theory of Simha-Somcynsky [10] may be used to arrive at values of dTt /dP = LlK / Lla.
As indicafed previously, eqs (5) are based on several linearizations .It is assumed that th e coeffi cients ai, ag, Kl and Kg are independent of pressure and temperature and that T g is a linear fun ction of pressure.Thus strictl y, the reference temperature in the glass as well as the initi al temperature in th e melt should be a ppropriately close to T g.Moreover, the pressure pi should be appropriately small.In the Appendix th e general relations hips are developed, based on the eq uations of state of the liquid and both glasses.
As an exampl e, integral relations are evaluated over the two paths s hown on fi gure l a for PV Ac, for which extensive data are available [1], a nd the Tait parameters are known [14] for the liquid and both glasses.The results are tabulated and compared with the corresponding linear approximations.

Application of Molecular Theory
Thli hole theory, which is used here to estimate the values of dTT /dP , is a corresponding states theory based on a lattice modef.The partition fun ction is defined in terms of a single ordering parameter, the hole frac tion h, which gives the ratio of the number of vacant to total sites, each of which may be occupied by a polymer segment.The corresponding states are given in terms of th e reduced (universal) variables where T* , P *, and V* are the scaling factors applicable to each pol ymer.Although these are defined explicitly by the theory, they are usually derived from a superposition of equilibrium PVT data along the master curves evaluated from the theory.For an illustration of this procedure, see Ref. [14].
The partition fun c tion Z is expre §sec! uniquely in terms of the three independent variables T, V, and h.From the thermodynamic definition P = -kT[af nZ(T, V, h) /aV]T :1 Differential scanning calorimetry and the equilibrium constraint (aZ / ah)T,V = 0 , the following equilibrium equations [10] are obtained, respectively: [2-1 /6y(yVtI /3 -1/3][1 -2-1/6y(yVt1l3]-1 (10) where y = 1 -h is the fraction of occupied s ites, and sand 3c are the number of segments pe r molecul e and the external degrees of freedom pe r molecul e, respective ly.As in previous work, we take s/3c = 1.Note that the term (s -1)/s in eq (10) approaches unity for large molecul es .
A basic assumption sufficient for the application of the hol e theory to our de nsification model is dTt / dP = (aT / aPh.g (11) Gee [15] has shown that such an equation is valid for a si ngle ordering parameter which is frozen in the glass .However, since h has been found to vary slightl y with temperature and pressure in the glass [16,17,18], eq (11) must be revaluated to assess its validity for the more general case.
Since there are three independent variables (in the general case, the derivatives with two fixed arguments (suscripts) are the same for liquid and glass.(For the glass it is understood that these derivatives pertain to constant P I and k.)The differences become

(aT / ap)" = -(ah/ap)r,{/(ah/ aT)p,{
= -(ahJaPJr.J( a /h/ aT) T,g, it follows that which, as seen from eq (13) , is tantamount to F T = F p. Accordingly , since eqs ( 12) and ( 13) coinc ide, th e validity of eq (12) is extend ed to a s ingle ord e ring paramete r whi c h need not be "frozen" in th e glass.
To our knowl edge both of th e above freezing fra ctions have been evaluated for onl y two systems, na mely poly(vinyl acetate ) [16] and se le nium [19].According to th e best a nalysis give n in ref. [16], F p = 0 .88and F T = 0.82 for which the ratio F p/F T = 1.07, which corresponds to a 7 pe rcent disc re panc y in eq (11) for poly(vinyl acetate).A s imilar conclus ion follows for Se.Since th e above analysis shows tha t F T = F p , these differe nces are tak en to be artifac ts resulting from nume ri cal inacc urac ies .The next step is the evaluation of (aT/aft )" at equilibrium .
From simultan eous numerical solutions of eqs (9 ) a nd (10), values of h = 1 -yare obta ined at a give n set of reduced te mperatures a nd pressure §.Fp r computationa l pUl~oses jt is conve /!i ent to re pl ace (aT/ap)" by th e ra tio -(ah/ ap)r/ (ah / aT)p.With constant inc reme nts I1 x(x = Tor P) , it is easily s hown fo r a quadrati c de pende nce of y on x th at (dy/dx)i = (YHI -Yi-l) / (2I1x).
Equation ( 16) may be re written (see for exampl e eq (14) in Ref. [14]) as -l1a[(aT/ ah)p X dh/dP] (16') wh ere th e total de riv a tive on the right hand s ide is to be tak e n along th e Tg(P) lin e. Provided the pressure coeffi cient of T 9 has bee n de te rmined with sufficie nt acc urac y, th ere appare nt ly is no nume ri cal advantage in using eq (16').
It is me ntioned in the last section that eq (7 ) a ppea rs to be a good approximati on for most polyme rs.Assuming thi s relation, we may estimate K' from volume-te mpe rature and heat capac ity data us ing the relation Olabisi and Simha [17] have shown for most polyme rs studied by them that the scaling [actor p* may be de te rmin ed from the other two by means of the empirical relation wh e re th e dimens ions a re K, bar, a nd c m 3 /g .Thus it appears fe as ibl e to estimate K' from appropri ate data at atmos phe ri c pressure onl y. Thi s poss ibility is tested later in thi s pa pe r.
According to Wunderli ch [20] it is poss ibl e to es timate I1C p a t Tg to within a bout ± 2J /(mol-K) by a ppl yin g th e " rul e of constant I1C p".The molecul a r re pea t units a re broken up into fund a me ntal units or " beads" whi c h loosen up in th e Tg process.Eac h bead is ass igned th e valu e 11. 3 J /(mol -K).
The contributions of the beads to I1C p a re ass umed to be additive.Accepting the va l idity of thi s rul e, it a ppears poss ;ble to obtain a crud e estimate of K' from hole theo ry appl ied to volume-te mpe rature measureme nts a lone.

. Data Sources
Although we refer usuall y to th e ori ginal sources, the re are coll ec tions of data on the pe rtinent q ua ntiti es in the lite rature , whi c h are some times c ited he re .Exte ns ive li sts of polymers and the ir va lues of l' 9 a ppear in re fs.[21 -23], whe re the last is restricted to fluorin e-containing sys te ms.
Tables of T 9 and l1a are included in Refs.[24][25][26], and T 9 and I1C p in Refs.[27] and [28].Reference [29], whi ch is occasionally cited here, contains a more critical evaluation of C p data on polymers for which the values on the same substance are often based on ave rages from different sources over wide ranges of temperature.Lists of polymers and their scaling factors based on the hole theory appear in Refs .[17] and [30]; however, p* is not available in the latte r.Pyrol ys is data on polymers are containe d in refs.[31 -34].These are useful to prevent degradation during the de ns ificati on process and to optimize the amount of densification .Finall y, ref. [35 ] gives an extensive list of refractive indi ces [or polymers.
The number of digits for the values given in the subsequent tables is not intended to be an indicati on of prec ision or accuracy.Usually thes e numbers correspond to those given by the data sources.It is our opinion that most of the entri es in these tables have more digits than can be jus tifi ed as significant.
Tabl e 1 gives the lists of polyme rs studied , abbreviati ons used here, and the ir values 0[1' g.In all tables the sequence is in order of increas ing T g.Table 2 gives the scaling factors based on the hole theory of Simha and Somcynsky [10].These are determined through superposition of experimental equilibrium data on each polymer with respect to the theoretical equation of state.In this work the scaling factors are used solely to estimate dTt / dP = t::.K/t::.afor each polymer using eq (15).When two nu~bers appear in the reference column (in table 2 only), the first applies to the data source, and the second to the work by which the scaling factors are evaluated.When only one number appears, the scaling factors are either evaluated in the reference given, or by us.Two values of p* for each polymer (or row) usually appear.The first of these (P*) is determined in the usual way through superposition as mentioned above.The second (Ptalc) is obtained from eq (18).When volume-temperature data are available at atmospheric pressure only, it is necessary to use eq (18) to estimate P*.With the exception of PDMSi, p* and Ptalc agree to within 17 percent with an 8 percent relative standard deviation of differences over 17 pairs.With PDMSi the disparity of 87 percent is outstanding, and it is to be noted that the reduced glass temperature lies significantly outside the range for which eq (18) was deduced.Similarly, V* and T* are obtained at considerably higher temperatures than those employed here.A decrease of T* by 7.2 percent and a concomitant decrease of V* by 2.4 percent over 100 K has been estimated [30] for this polymer.Accordingly, the scaling factors cannot be assigned significant constant values over the experimental range.

Scaling Factors
In order to obtain some measure of the uncertainty in the scaling factors , several data sources on each polymer are sometimes included.

Densification Rates from PVT Data
Table 3 gives the results of calculations of the densification rates from PVT data without recourse to molecular theory.K~ is determined from the definition: (4') which is identical to eq (4) setting t::.P = P', except that V A replaced v.o.The difference between the values of K; determined from eqs (4) and (4') are insignificant in comparison with experimental uncertainty.K~ is determined from eq (5b).Note that there are only two polymers, PV Ac and PaMS for which we found sufficient information to determine both K; and K~.Although the two methods are not necessarily fully equivalent because of the assumptions used to derive eqs (5), the agreement in both cases is good.In instances of more than one set of values per polymer, it is clear that the deviations in dTt / dP have the largest effect on the uncertainty of K;.Th:se apparent discrepancies are usually consistent with the differences in t::.K.With polystyrene the maximum deviation in dTt / dP is 38 percent compared with those for dT g/dP, 23 perc~nt and t::.a, 10 percent.Since K~ involves the difference between the two transition rates, its maximum deviation is magnified to 56 percent with its relative standard deviation over the five values being 26 percent.
It is interesting to note that the direct method giving K; which one might expect to be more reliable, yields values for which the maximum deviation (for polystyrene) is 86 percent with relative standard deviation over seven values being 35 percent.The ratio of average values, KVK~, is 1.6.These discrepancies are a measure of the difficulties in obtaining reliable PVT data on glassy polymers.
In many instances the values of t::.a at T g and the required transition rates are not tabulated in the data sources and, therefore, had to be evaluated.The accuracy of these evalua-tions may be conside rab ly limited whe n the data are presented in graphical form only.In ref. [47] the values of f:J.K are determined by a different definition from the one used by us.In our definiti on Kg is take n to be an isochronal (nonrelaxing) fun c ti on of te mpe rature and pressure and therefore must be derived from data at temperatures below (or pressures above) the glass transition region.f:J.K at Tg is then obtained by extrapolation.This, apparently, was not done by Hell wege et al. [47], at least over the appropriate temperature range for the data to be effectively isochronal.The distinction between the two definitions is clearly illustrated by Boyer [57].Our larger values of f:J.K are determined from the Tait parameters given in ref. [53], which apply to the data of Hellwege et al. [47].Note that the values of d7i /dP from reevaluating [53] their data are in good agreemenf with most of the others on the same polymers except for PMMA.This discrepancy would be increased by using their value of f:J.K given in ref. [47], along with poorer overall agreement with the other two polyme rs .In ref. [1] the transi ti on rates are given as tange nt values along the trans ition lines at each experimental pressure.Here, we use th e secant values dTg/ dP a nd d7i / dP betw ee n o and 800 bar.This procedure gives average gv alues a nd is more cons iste nt with oth e r treatm ents .

drJ/dP
As stated in section 3 the trans ition rate dTt / dP = f:J.K/ f:J.a applicable to the constant formation hi sfory, may be estimated from the value of (ar/DP)" at T g.Table 4 summarizes the results of these calculations.In all cases exce pt for PDMSi and PIB the data encompass T g.With th ese two polymers the first reference for each applies to the source of data at atmospheric pressure, and the second , at elevated pressures.Equation ( 14) and the scaling temperatures and pressures as applied to eq (15) provide the requisite information.The distinction between the values of (DT/DP)" in columns A a nd B is that they correspond to p* and Ptalc, respectively, in table 2. Values of dTr / dP are included for co mpa ri son with those of (DT/DP)" in ~ases wh ere th ere is sufficient ex pe rime ntal information.
With natural rubber (NR) f:J.K is de termine d from dyn ami c compressibility data [60].This involves the measureme nt of th e ad iaba ti c compress ibility in a hydrostati c stress fi e ld alte rn a tin g at low audio freque nc ies.The low-a nd hi gh frequency limiting compressibi liti es a re conv erted from adiabatic to isothe rm al conditions, prov iding th e difference f:J.K.The fa ct th at f:J.K is dete rmined at about 20 K a bove normal T g is expected to have no ap prec iabl e effec t.
Except for PDMS i, the cOiTes ponding va lu es of (DT/DP)" are nearly the same in co lumns A a nd B. The di sc re pa ncy for PDMSi is exp la ined by th e fac tors me nti oned earli e r. (See sec .4.2.).Excluding this polymer, the rela tive standa rd de viation of the differences betwee n co rrespo nding va lues in these column s is 6 pe rce nt, whi c h is cons ide red to be good ag reement.For co mpa ri son with ex pe rim e nt, th e residual sta ndard de viati on betw een co rresponding values of (DT/DP)" (column A) and d'1 / dP is 18 pe rce nt.Poor agreeme nt is noted for PnMBa , a nd two sa mpl es of a -PMM A.
Ove r a s in gle substa nce, for exampl e po lys tyre ne where we have four sets of values, the re lativ e s tanda rd deviation s with respec t to the averages for (DT/DP)" (column A) and dTt /dP a re 6 a nd 15 perce nt, res pective ly.Tha t for the differ~nces between corresponding va lues of these quantiti es is 11 percent.Thus, based on th ese simpl e stati stics , th e most serious limitation is not the inadequacy of the theolY, but the uncertainty in the experimental de termina tion of dTt / dP.The agreement be tween values of d1j / dP from differ~nt investigators is even worse for a-PMMA .

Application of Hole Theory
After determining the values of (ar/DP)" for each substance, K' may be estimated by eq (16), where dTg/dP may be determined by means of PVT data, dynamic measureme nts at elevated pressures, or heat capacity and the rm a l ex pa ns ion data, both at atmospheric pressure.

PVT Data
Since PVT measurements a re often made by th e variabl e formation history only (for exampl e, pol ypropyle ne, ref. [40]), there is insufficient information to determine f:J.K, a nd hence dry / dP, to be applied!oeq (5b).Acco rdingly, this qu antity is re placed by T*( ar/DP)wP* leading to eq (16).The results of these estimates are given in tabl e 5, wh e re K' co rres ponds to K' 2 in table 3.In all cases (arjap)h is taken from column A TABLE 5. Densification rate caLcuLations using PVT data and hoLe theory of table 4. Since the expression for K' involves the difference be tween two transition rates, its value is very sensitive to (arj ap)h.This effect is reflected in the large standard deviation, 2. 5 Mbar~ \ with a relative value of 28 percent, for the differences over 14 pairs of corresponding values of K' 2 and K' in tables 3 and 5.The values of PaMS are not included in this calculation because there are insufficie nt data in ref. [56] to determine the scaling fa ctors applicabl e to this paltic ular sample.Based on the fact that th e s tanda rd de viation of K' 2 for polystyrene (table 3) over five values is 2.8 Mbar ~ I corresponding to 26 pe rc e nt, the overall 28 percent value above appears to be dominated by experimental uncertainty.

Dynamic Data
From the assumption that the value of dT JdP, approximates that of (aTjap)w, where w is the angular freque nc y, freque ncy-tern perature-pressure superposition of dynamic data, including dielectric and ultraso nic, may be used to determine dT JdP appearing in eq (16).The validity of thi s assumption is, of course , subject to the co ndition th a t (aT j ap)w for the T g -process is essentially independent of frequency.(Numerical comparisons be tween differe nt ex pe rime ntal transition rates are made below .) The results of these calculations are given in table 6.Since values of (aT jap)h in column A of table 4 involve fewer assumptions than those in column B, the form e r are used whe re there is a choice.The standard deviation of the diffe rences of K' over eight pairs, where there are values on the

Heat Capacity and Thermal Expansion
The estimation of K' from heat capacity and thermal expansion data is based on the apparent validity of the second Ehrenfest equation [eq (7)].(This relation is tested in the next section.)The results of the calculations based on eq (17) are summarized in tabl e 7. In this case the statistics may not be meaningful because there are only five values of K' whic h correspond to those in table 3 including Kl for Pcarb.PaMS is excluded for the reason given above.The standard deviation of the differences is 1.4 Mbar~ 1 or 17 pe rcent, whi c h is somewhat less than the experimental uncertainty (26 pe rce nt) based on polysytre ne data (table 3).In vi e w of the high experimental unce rtainty for all me thods , this me thod of estimating K' a ppears to be reliable, except for PDMSi and PaMS.
In ref. [12] a negative value of K' for PDMSi [based on eq (17)] is reported.This is a surprising, but not necessarily an incorrect result.The analysis of this polymer is hindered by the lack of good thermal ex pansion data through T g, largely a consequence of the low temperatures required, and the strong tende ncy for this polymer to crystallize.The negative value of K' is obtained by using the value of ~a = 10.28K ~ 1 from table I of ref. [26].This value is based on the linear thermal ex pansion data of Weir, Leser, and Wood [58].After a thorough examination of their results and consultation with Dr. Wood, it was dec ided that the temperature range for which Val was evaluated is too small and too remote from T g to evaluate ~a at T g.In order to obtain what we consider to be the best available estimate of ~a, we used the value of Vag = 2.7 X 1O~4 cm 3 j(g-K) from ref. [58], and al = 8.7 X 1O~4 K ~1 and V g = 0.904 cm 3 jg from the de nsity-temperature equation of Shih a nd Flory [36].A lthough thi s eq uation is derived from data at te mpe ratures we ll above T g , its nearl y linear response appare ntl y allows valid extrapolati on to mu ch lower temperat ures.The value V gal = 7.9 X 10-4 c m 3 /(g-K) is slightly less than the average, 8.7 x 10-\ of th e oth e rs for this polymer in tabl e I of ref. [26] which are obtained from differe nt sources of data at higher temperat ures not encompassing T g.Also the ex trapola ted value of V g = 0.904 cm 3 /g above essentiall y coi nc ides with 0.905 in ref. [26).The revised the rmal expansion values give the positive value of K' shown in table 7.

Comparison of Experimental Transition Rates
In tables 6 and 7 the assumptions that dT g/dP could be re placed by (OF/ap)w or TV l1a/I1C p, respec tively, a re employed.In table 8 values of these quantities are compared for each polyme r.A s imilar table was prepared by O'Re ill y [13] in 1962 for glass-forming liquids not restri cted to polymers.
Values of I1K/l1a are also included here for comparison; however, agree me nt with dT g/dP is not expected since th e validity of the inequality appears to be quite strong and general.In most instances, agreement between dT g/dP, (ar / ap )w, and TV q./ I1C p seems to h ~K = 1.2 X 10-5 bar-I determined from dynamic compress ibilit y.
C All given determinations on same sample.
d Measurements by J. J. Weeks reported in Re f. [16].
be within experimental error.Small diffe re nces may be a nti cipated because the conditions under whi c h these quantiti es are evaluated may be vastly diffe re nt.
According to these results the Prigogine-De fay ra ti o is essentially unity for natural mbber and PaMS.Unfortunately, we do not have a PVT value of dT gldP for the former to test the validity of the Ehrenfest equations [eqs (6) and ( 7)].
With PaMS it would appear that although neither of the Ehrenfest equations is obeyed, the Prigogine-Defay ratio is still unity, which is an atypical result.This implies that d7i /dP for volume and entropy are equal, but dTgldP is dis~inct.However, since data in tabular, or even graphical form, are not included in ref. [56], evaluation of these numbers cannot be scmtinized.Poor agreement for PVC in all cases is apparent; however, this may be a result of sample differences including the degree of crystallinity which is difficult to control in this polymer.Also poor agreement is noted for a-PMMA of ref. [9], where both quantities are obtained from the same sample.On the other hand, the data in the row above on the same polymer reveal good agreement including that with dT gldP (PVT) of ref. [9].In all cases agreement is very good for polystyrene.
These results indicate that dTgldP = (aTlap)w is a valid relation and dT JdP = TV t:.al!JCP seems to hold most of the time.The validity of the first may be argued on a qualitative phenomenological basis (see sec.2.) The second relation is evaluated at atmospheric pressure only.There is no apparent reason to assume that the approximation will be as good at elevated pressures.
In section 4.3 we mentioned that PVT data on PnBMA suggest that the first Ehrenfest equation [eq (6b)] is a good approximation for this polymer.This result is tantamount to essentially no densification.(See table 3.) Unfortunately, we have no heat capacity values for this polymer, which are needed to check the second Ehrenfest equation [eq (7)].In section 3 we noted the possibility of using the "mle of constant t:.C p" [20] to estimate the heat capacity difference at T g. (For a comparison of experimental and "bead" values of t:.C p on polymers, see ref. [28].) For PnBMA the molecular weight of the polymeric repeat unit is 142.2 g/moJ.Assigning one bead to each of the two carbon backbones, and one to the oxygen atom, we obtain a total of three beads, which for 11.3 J/{mol-K-bead) gives t:.C p = 0.24 J/(g-K).Taking this value along with those for t:.a and T g from tables 3 and 4, respectively, and V g = 0.946 cm 3 /g from ref. [43], we obtain TV t:.al t:.C p = 20 K/kbar, which is in good agreement with 20.4 in table 3. Thus both of the Ehrenfest equations appear to be fairly good approximations for this polymer, along with a corresponding Prigogine-Defay ratio of nearly unity.(The value 1.2 is obtained for PnMBA.The average value obtained from table 8, exclusive of NR and PaMS, which were treated separately, is 2.1.)These results imply that both the density and entropy of PnBMA are essentially independent of formation pressure, at least at low pressures.

Pressure Dependence of K' and Limitations Imposed by Chemical Instability
The previous discussions in this paper pertain to the initial values of K' or at least at very low formation pressures.Most of these are either tangent values at p' = 0 (atmospheric pressure) or secant values obtained from P I = 1 kbar or less.
There are data in the literature, however, which include densifications obtained at different formation pressures.
There are two important physical considerations in optimizing the procedure to obtain "permanent," densified glasses.The first and more obvious, is to select and maintain the temperature of depressurization at temperatures sufficiently below T g.It is clear that the ambient conditions must be such as to maintain stmctural relaxation times which are large in comparison to the desired "lifetime" of the glass.Accordingly, high T g substances are preferable for room temperature stability.The second is to choose To, the temperature of isothermal pressurization, large enough that the equilibrium melt is always maintained during pressurization.Stated alternatively, the inequality (19) must be approximately satisfied, as illustrated in figure 1.This condition implies that the effective time of the pressurization process must be large in comparison to the stmctural relaxation time at the final pressure P I.In cases w here To < Tg(p') there will be a much smaller contribution to the densification process when the condition To = T g(p) is approximated and exceeded during pressurization.This situation is revealed by a leveling off in the volume as illustrated schematically in figure 4, where volume changes are plotted with respect to formation pressure at different pressurization temperatures.The densification is expected to be independent of To at low pressures, when ineq ( 19) is satisfied, as is revealed by the coalescence of these curves with decreasing P'.Such a coalescence is not expected, however, when To < T g as is illustrated by the data of Shishkin [54] on polystyrene.In figure 4   To on the densification process.
The dashed line is the envelope approached at large temperatures.
One of the better experimental examples which illustrates the behavior shown in figure 4 is provided by the data of Shish kin on PMMA and PS.Formation pressures up to four kbar are applied; but not all of the pressurization temperatures are above T g(p l ).At the lower pressures, K' increases with P' as is indicated by the increasing slopes of Shishkin's volume-formation pressure curves, and as is shown in figure 4.This is th e oppos ite of the trend for the isoth ermal compressibility K , whi ch decreases with increas ing pressure (see for examp le re fs .[12] and [74]).The data of Sh ish kin , as we ll as those of Kimmel and Uhlmann [75] on PMMA , show that some densifi cation is possible with pressurizati on temperatures below T g , but th e effec t is diminished as th e diffe re nce between th ese two temperatures is increased.Other exampl es illustratin g the de pe ndence of densification on formation pressure are refs.[6], [9] and [55] on polystyre ne, and [56] on PaMS.Yourtee and Cooper [6] observe a velY slight decrease in the densification rate with formation pressure for polystyrene over a 6 kbar range.For the same pol ymer, Weitz and Wunderlich [9] find a much larger dependence wi th the same trend , where the d ensity graduall y becomes nearly constant at 4 kbar.These trends are contrary to the marked increase in the densification rate with formation pressure observed by Shishkin on PS and PMMA a nd Ichihara e t a!.[56] on PaMS.K ' does not necessarily have to te nd to zero for the vo lume to be non-negative at large formation pressures.Using our definition of K' , the de nsified volume te nds to zero at a rbitrarily large formation pressures when K' is a positive constant.
According to th e ex perime nts of We itz and Wunde rli ch on polystyrene, the re is a monotonic increase in density at a decreasing rate which the density seems to level off at 4 kbar.Thus, beyond this point th e formation pressure would have no effect on the densification process.On th e oth er ha nd , with most of th e other investiga tions mentioned above, including those on polystyre ne, it would appear that che mi cal stability is the limitin g factor.Whether the reacti on rate constant of a given rate process increases or decreases with pressure depends upon the sign of its corresponding ac tivation volume [76].In most cases it is expected that the total activation volume will be pos itive with a corTesponding increase in the effective decomposition tempe rature with in creas ing pressure.Thi s behavior may be complicated , however, by th e diffe re nt temperature and pressure dependencies of th e various decomposition modes, and, possibly by the ini tiation of new ones at elevated pressures.
The important consideration here is whether, 0 1' not, th e decomposition tempe rature and T g -pressure curves come sufficiently close at any point to limit the densification process .For example, with polytetrafluoroeth yle ne the inc rease of decomposition temperature with pressure is only about 3.5 Kj kbar [77].Although this rate is small, the decomposition temperature is sufficiently remote from the observed phase transitions, since its initial (atmospheric pressure) value is about 700°C.In addition, the melting and decomposition curves diverge with increasing pressure over the experimental range of 28 kbar, investigated so far.
In cases wh e re the decomposition te mpe rature Td increases with pressure, To should also be a ll owed to increase with pressure to optimize the densifi cati on.With pol y tetrafluoroethylene this process would appear to co nti nu e wi th out bound because of the observed divergence menti oned above.
In instances where T d and T g conve rge or intersect a t a finite pressure, the densification would be essenti a ll y I imited by the effective intersection temperature as illustrated in fi gure 5. Except for the polymer me ntioned above, press ure dependent pyrolysis data are a ppare ntl y non-ex iste nt in the literature 4 •  The results in tabl e 9 s umma rize an attempt to es timate the optimum densification on a few polymers beyo nd whi c h the rmal decomposi ti on wou ld occu r.In th e a bse nce of re i iabl e pyrolysis data at elevated pressures, we wi ll estimate optimum densifi cati on by co mm enc ing isothe rma l pressurization at To = T d.Since in most cases T d is expected to increase wi th pressure, thi s p rocedure should underestimate th e max imum densificati on as illustrated by th e lowe r va lue of P ' max obtained by the dashed line path in figure 5.In table 9 T d is taken arbitrarily at the value for which the initial reaction rate constant k = 1 % /hr, applicable to the total degredation process .Assuming Arrhenius behavior T d may be calc ul ated from the relation
T is arbitrarly chosen from the closest data point to T d, which in all cases, but one (PV Ac), invol ves extrapolation.These decomposition temperatures correspond to those given in table 7 of ref. [34], except th e latter apparently apply to k = 1 percent/min a nd, accordingly, are la rger.The ceiling temperatures in the same table, which a pply to the propagation mode at eq uilibrium , are appare ntl y not relevant to the densification process.T g, dT g/dP, and K' a re selected from previous tables in this paper.P'max, the pressure corresponding to the onset of pyrol ys is at To = Td and -(dV/ V)max, the corresponding maximum de nsification , are obtained from the s impl e relations P'max = [Td -TuCO)] / (dTgl dP) -(dV I V)max = K' X P'max' From these results it appears that P' max varies inversely with T g; however, no tre nd is appare nt for (dV!V)max.

Dependence of Refractive Index on Densification
A reliable estimate of the change of refractive index on de nsification s hould be obtained by means of the Lorentz-Lorenz equation, (20) wh ere n is the index of refraction and p the density.K depends upon the polarizability, which is expected to be essentiall y independent of formation pressure, or alternatively, the density at constant temperature and pressure.The relative change of index of refraction wi th formation pressure, fj' = (lin) (an/ap)T,p, is obtained explicitly by differentiation of eq (20), viz, Table 10 presents the results for polymers for which values for nD (sodium D line) are available from ref. [35] with K' selected from our tabl es.The nD values are converted to those at T g by means of the temperature coeffi c ien ts given in ref.
[84].As seen from the table, these corrections are insignificant.Since all of these val ues range betwee n 1.48 and 1. 58, a very slight (10 percent) error will be incorporated in fj' by ta kin g the function fen) = (1/6n 2 )(n 2 -l)(n 2 + 2) as a constant, as revealed by the table.Accordingl y, in vicw of the large experimental uncertainties in K' (35 percent for polystyrene), the addi tional uncertainties obtained on replacing eg (2 1) by the approximation fj' = O.4K' are slight.The values of 0' in the table however, are calculated from eq (21).We do not nave any direct ex perimental data giving the dependence of the index of refrac tion on formation pressure.These evaluation s have potential application in optimization or adjustment of the refractive indices of plastic lenses by appropriately setting the mold ing pressure.The values in the last column in table 10 give the relative percent changes (Lln /n) resulting from a moding pressure of 10 kbar.For PS a nd PMMA , which are common constituents for plastic fenses, n would c hange by 4 and 3 percent, respectively.Howe ver, it was estimated in the last section that thermal decomposition of these polymers would limit the pressurization to 7 and 8 kbar, respectively.In these analyses isothermal pressuri zation is considered at the decomposition temperature.If this temperature increases with pressure as indicated by ref. [78] for PS and PMMA, an additional increase in their refractive indices could be obtained by appropriately increasing the temperature during pressurization.

Conclusion
Several methods have been evaluated to estimate the densification rate, K', applicable to glass formation by iso baric cooling at constant rate.Other than the direct measurement of the volume difference in the glass, K' is always computed from an expression involving the difference between two transition rates, dTgl dP and dTt I dP.The hole theory is s hown to be sufficientl y accurattf in estimating dTt I dP for the 23 polymers evaluated except for possibly thC:Se of dimethyl siloxane and a-methyl styrene.With these it is not clear whether the discrepancies res ult from ex perimental error or lack of generality in the application of the theory .
Although dT g/dP is onl y evaluated experimenta lly , there are independent alternatives .The simplest of these involves the differences between thermal expansions and heat capacities at T g for liquid and glass at atmospheric pressure only.
The principal problem in the estimation of densificati on using these procedures appears to be the large amou nt of experimental uncertainty in all of the relevant quantities, in particular, the compressibility.Since the expression for K' involves the difference between two quantities of similar magnitude, eve n small experime ntal errors may have a pronounced e ffec t.Accordingly, it is diffi cult to assess th e relative merits of th e diffe re nt meth ods e mployed here, including th e app li ca ti on of th e hol e th eory.
The res ults of these a nalyses appear to have practical applications .Densifying glasses produces a ha rd ening e ffect as revealed by an inc rease in moduli.Howeve r, th ese effec ts do not appear to be as pronounced, in parti c ul ar viscosity or relaxation time, as those obtained at the sa me volume, te mperature, and pressure in the glass by commensurately dec reasing the cooling rate at atmosphe ri c pressure.Thi s procedure howe ver is usually not practi cal because of the large times required for glass formation.According to one investigation it is possible to optimize the ultimate properties through the appropriate adjustment of the formation or molding pressure.More work is necessary to establish the generality of this result and to determine the formation pressures for maximum yield s tress .Moreover, th e re lation between th e refractive index a nd densificati on quantity presented could be used to quantitatively regul ate the refractive index of lenses through a ppropriate adjustment of th e molding pressure.The maximum value would ap pear to be limited by che mi cal insta bil ity at the hi gh tempe ra tures necessary to exceed l' g(P), which increases with pressure.The simple relation give n does not include the inf1uence of clensification on optical di spersion.Again , ex pe rime ntal work is req uired to assess th e validity of our estimates and the poss ibl e inf1ue nce of densification on dis persion .Thi s work was supporte d in part by the National Science Foundation under Grant DMR 75-15401.

Appendix
As indi cated earli er, eqs (5) in the text involve a linearization of pertinent quantities and proximity to the transition line Tg(P) in figure 3. The experimental data found in the literature often do not satisfy th ese conditions.Hence we reconsider here the processes depicted in figure 3, by replacing th e simplifications adopted earli er by the more general form.This will not only permit the prediction of densification effects under more extreme conditions, but also allow us to gage the quantitative validity of the linearization.Clearly, a knowledge of the equations of state is required for an explicit evaluation, but is certainly not available for the wide range of systems discussed here .However we shall be able to present typical numerical illustrations using PV Ac where appropriate data exist [1 , 14].
From the definition s of the coeffi cients a and K it follows that: where the subscript indicates an initial value.However, it will be accurate enough to omit second and higher powers in the expansions of the ex ponentials.Considering an average, temperature independe nt value for the liquid, (a) = 8 x 10-4 K-1, and a tempemture interval of 100 degrees, we obtain for the integral a value of 8 X 10-2.Thus the quadratic term changes the l::>tal result by 0.34 percent.The values were chosen so as to magnify th e effec t.For th e glass, th e approximati on will be even more adequate .With a press ure diffe rence of 2 kbar and (K) = 4 X 10-5 bar-r , the magnitude of the relative volume c hange is th e same and ide nti ca l conc l usions are obtained as for the a-term.
De noting th e initial temperature in the me lt as To and th e fin al te mpe rature in the glass as Tf, with the initi a l press ure tak e n as zero, we obtain instead of eq (3), wh e n th e pressure de pendence of the a's and the temperature de pende nce of the K' s are taken into account , the followin g ex press ion : (A-I) wh ere th e subsc ripts b a nd c pertain to the low and hi gh pressure glasses respec tively.This c hoice conforms wi th the nome nc lature used in refs.[1] and [14].In th e lin ea rized de rivation, a and K for both I iquid and glass are take n to be constants.Accordingly, th e re is no di stin cti on be tw ee n th e values of ag,b and ag,c, and, simil a rly , Ky ,b and K y •c.
To proceed furth e r, we make use of a n ex pli cit equation of state.It is most conve ni e nt to e mpl oy the ex te ns ive ly tested Ta it relation s for both melt a nd glass.To recapitulate the pertinent equa tion ( 14), [50]:  where the first two terms will predominate.We now proceed to evaluate the integral s in eq (A-3) which are identified as follows: (A-4) where the terms to the right of the arrows are the corresponding linear approximations used in eqs (5) in the tex t.
Since C = 0.0894, the compressibility may be written in good approximation , and consisten t with the expansion of the ex ponential s above, as C/(B + P).Thus we find for 11.
12 and 13 are evaluated by expressing the atmospheric pressure volume as Vo ~ Ao + BoT + CJ2 for whic h a = (Bo + 2C oT)jV av is a good approximation.Vav is take n as the average of the two bounds.Finally the general integral corresponding to 14 and 15 is J K(T, P) dT = (C/P) {T + (l/b) en [P + a exp( -bT)]} + I(P).
Using the parame te rs for PV Ac given in tables 1 and 2 of ref. [14], the values of the integrals and their linearized counterparts are s ummarized in tabl e A-I.From ref. [1] T y(0) = 30.7 °C and Tg(P') = 48.00c.To and Tfare taken to be 90 and 0 0c.These two temperatures are conside red to b e s ufficiently re mote from T g(P) , to be characteristic of the equilibrium and glassy states, respectively.The total relative volume differences are given at the bottom of the table followed by the corresponding dens ification rates.The differe nce between the two values of K' amounts to about 4 percent, whic h is quite satisfac tory, since the experimental error on this quantity appears to be considerably larger.This good agreement seen in table A-I arises however from a cancellation of approximation errors.Moreover, it is gratifying, that the value K' = 8.8 Mbar-1 , based on the Tait equation is essentiall y identical to that obtained by directly measuring the volume difference.(See sec.4 .3.)This illustra tes the satisfa ctory performance of the analytical expressions in representing the experime ntal data.

FIGURE 1 .
FIGURE 1. Schematic illustration of two methods used to obtain densified glasses.
(a) Densification by elevating the fonnatioll pressure at the sa me cool ing rale k j • (h) The SIII11(' densification is obtained by commensurately decreasi ng the cool ing ru le al llimospheric p reSsure.

1
Figures in brac kets indicate literature references at the end of this paper.2 For conversion to 51 units, I MPa = to bar.

FIGURE 2 .
FIGURE 2. Schematic illustration of two thermodynamic histories used to form Idasses.
FIG URE 3. Schematic ilL ustration of the procedure u.sed to derive the densification equation {S ee eqs (3) and (4)} , and the distinction between T.(P) a nd Tt(p , P').

9
the dashed line represents the extension of the envelope established from arbitrarily large values of To.

FIGURE 4 .
FIGURE 4. Illustration of the influence of the pressurization temperature

FIGURE 5 .
FIGURE 5. Schematic illustration of temperature-pressure history used to optimize the densification process before the onset of pyrolysis.

TABLE 1 .
List of polymers studied, abbreviations, and glass temperatures a 55 percent Styrene.

TABLE 2 .
Polymer scaling factors [53]* and V* only are determined in this reference.bVolume-temperaturedataat atmospheric pressure taken from previous listing.c55percent Styrene.dForadditionalcomments of interpretation on these experimental data and evaluation of Tait parameters, see Ref.[53].

TABLE 3 .
1 in tabl e 3 applies to PnBMA.Suc h a s ma ll value implies that th e firs t Ehrenfest equation is a good approximation for this polymer.[See eqs (5c) and (6b)].Densification rate calculations at atmospheric pressure fro m PVT data
Mbar~ I or 30 percent, which is about the same as the experimental uncertainty given above (2.8Mbar ~ I or 26 percent) for polystyrene.This value is also about the same as the 28 percent value given for the PVT data even though data on different s ubstances are involved.It is possible, however, that dT gjdP values determined from dynamic data, in particular dielectric , whe re high resolution is obtained, are more reliable than PVT values.T g determinations from PVT data usually involve extrapolations which are not used in the superpositi on of dynamic data.

TABLE 6 .
Den.sification rate calculation.sfrom dynamic data.at elevated pressures and hole theory

TABLE 7 .
Densification rate calculations from thermal expansion and heat capacity data.and hole theory a adT .ldP= T.V ~al ~C 1" h Partially crystall ine sa mpl e. C All quantities derived from the sa me sampl e.

TABLE 8 . Comparison of experirnenlal transition rales a
Re f.

TABLE 9 .
Estimation of maximum densificationfrom pyrolysis data 1 a Stated to be unstable at temperatures above 463K In Ref.[81].DUring sampl e pre paratIOn [11 slight d,scolorallOn was obse rved afrer heating ove nllght In a vacuum at 403 K .

TABLE 10 .
Estimation of change in refractive index from densification rate b Taken from Ref. [84J.