Thermodynamic equations of state of polymers and conversion processing

Transcription

1 Polimery, No.,,. 66 hermodynamic equations of state of olymers and conversion rocessing B. Kowalska Selected from International Polymer Science and echnology, 9, No.,, reference P //66; transl. serial no ranslation submitted by E.A. Inglis hermal rocesses, in addition to rheological rocesses, condition the behaviour of a lastic in rocessing equiment and the roerties and structure of the roducts. he thermal state of a lastic is determined by the following factors: the ressure, secific volume and temerature. An imortant consideration during rocessing is the deendence of the secific volume on temerature at atmosheric ressure (Figure ). he relationshi between the ressure, volume and temerature is called the thermodynamic equation of state, or more simly the equation of state. Under equilibrium conditions the equation of state of a single-hase system is given by the equation () (ref. ): F (,, v) = () he simlest thermodynamic equation of state is the equation for a erfect gas in the form = nr () where R is the universal gas constant, [R = 84 J/ (kmol.k], and n is the number of moles of the substance. his equation assumes that the gas molecules are oint masses and hence have no secific volume, and do not interact with each other outside the moment of elastic imact. However, the molecules of real gases do have a secific volume, hence the effective accessible volume for movements of the molecules is less than the total volume of the system. he mutual attraction of the molecules causes a reduction in ressure, so that the molecules not interacting with each other move to the walls of the container in which the gas is located. In order to allow for the secific volume of molecules Van der Waals introduced a correction for the volume of the gas and reduced it to an active value, by subtracting the so-called co-volume b, equal to the sum of elementary sheres around all of the molecules. In the case of sherical molecules b is equal to four times their volume. he correction for ressure, called the cohesion ressure is according to Van der Waals roortional both to the number of attracted and attracting molecules, and thus is roortional to the square of the gas density ρ (ref. ); this relationshi is shown in equation (): a = aρ = () v where a is a coefficient of roortionality indeendent of the arameters of state. Because of this instead of equation () in its simlest form ( = R) we obtain: a + ( b) = R (4) his is an algebraic equation of the third degree with resect to ; after transformation it takes the form: b + R a ab + = In the analysis of hysical henomena it is desirable to emloy dimensionless magnitudes since in this way we can avoid the effect of the system of units adoted. For the thermodynamic dimensionless arameters we can take the ratios of arameters to their values in the critical state (also referred to as the reducing arameters, indicated below by an asterisk). In this way we obtain reduced arameters (indicated by the symbol ~) (refs. 4 and ): () /76 International Polymer Science and echnology, Vol. 9, No. 7,

2 ; ; = = = On introducing the reduced arameters into the Van der Waals equation we obtain the reduced equation of state in dimensionless form containing only universal constants (without material constants): (6) + ( ) = 8 (7) Equation (7) itself and any other equation of state in dimensionless form is a criterion of thermodynamic similarity; it is called an equation of corresonding states (ref. ). Different substances are in corresonding states when their reduced arameters have the same numerical values. he critical oints of each substance are corresonding states. From this it emerges that if any two fluids have two identical reduced arameters, then their remaining reduced arameter has the same value. he use of the rincile of corresonding states in technical calculations is of great imortance. According to the above rincile the relationshi ( ) = (8) F,, is the most universal equation of state for real gases and liquids. However, it is found from exerimental results (ref. ) that the state of substances being comared cannot always be described using only three reduced arameters. he theory has therefore been exanded by introducing a fourth arameter, the so-called determining criterion A. hus: ( ) = (9) F,,, A he value of A, which characterises the individual roerties of bodies, can be defined in various ways. Most commonly the criterion is the coefficient of comression of a substance at the critical oint (ref. ): Z = R () At a later stage we shall show how the -- relationshis can be used to further discover the henomena occurring during rocessing and to determine the connections between rocessing factors. USE OF HE HERMODYNAMIC EQUAION OF SAE IN HE CONVERSION OF POLYMERS he use of the equation of state in the conversion of olymers will be characterised using as an examle one of the commonest rocessing methods, namely injection moulding. he main factors in injection moulding are temerature, ressure and time. hese may vary widely. Variable external ressure affects e.g. the density of the lastic and the shae and dimensions of the moulding. By using a redetermined rogram of change in external ressure many roerties of the moulding are otimised (refs. 6 ). Simultaneous recording of ressure and temerature at the same site in the moulding cavity makes it ossible to lot their changes on a -- grah (Figures and ). he theoretical curve of the rocess (Figure ) marks out simle lines: the isotherm of ressure A, the isobar corresonding to the ressure hase B and the isochore of cooling C (refs. and ). Figure. -n- grahs reresenting a theoretical injection moulding cycle for olystyrene; isobars corresonding to ressure (in MPa): a., b, c 4, d 6, f 6 (for further exlanations see test) (refs. 7 and 8) Figure. -n- grah of a real injection moulding cycle for olystyrene; for meaning of isobar symbols see Figure ) (refs. 7 and 8) International Polymer Science and echnology, Vol. 9, No. 7, /77

3 At oint on the isotherm A the lastic reaches the ressure sensor. Point denotes comlete volume filling of the moulding cavity under low ressure conditions. Raid comression of the liquid lastic in the cavity is shown by lines. he lastic comressed to a ressure with a value defined by the isobar assing through oint and cooled remains fixed at the same isobar until it reaches a temerature at which solidification of the lastic at the narrowest oint occurs (oint ). his is a very imortant oint in the injection moulding rocess, which goes on sontaneously to the stage when he ressure of the lastic reaches a value defined by the isobar as. MPa (oint 4). Further cooling of the lastic causes a decrease in its volume and hence the occurrence of rocessing shrinkage. he actual curve of the injection moulding rocess (Figure ) deviated from the theoretical assumtions. Obtaining a constant value for shrinkage ( s ) is difficult because of the variation in osition of the oint where the lastic solidifies at the narrow oint and because of later uncontrolled changes in the secific volume (ref. ). he -- relationshi can be used to analyse the rocesses taking lace in the moulding cavity during different tyes of injection moulding, such as injectioncomression moulding (refs. and 4). his method of moulding has come to be used for making high-quality mouldings (e.g. comonents of otical devices, CDs). he volume rocessing shrinkage of the moulding when cooled in a tightly closed mould is comensated by a reduction in the dimensions of the mould cavity during the comression hase. his rocess requires two-stage closing of the mould and an additional comression force. he injection and comression hase of the mould are shown in Figure. he rocessing shrinkage of the mouldings accomanying comression can be determined using -- grahs (Figure 4). Figure 4. -n- grah for injection-comression moulding of olystyrene; designation of isobars as in Figure (for further exlanations see test) (ref. 9) Point in Figure 4 corresonds to the filling of the cavity. During flow of the lastic a slight ressure is maintained which is just higher than atmosheric ressure (oint ) until the mould is sontaneously sealed (oint ). hen the volume of the mould cavity is reduced as a result of the ressure, which causes a rise in ressure in the mould (oint 4). he conclusion of cooling of the moulding (oint ) occurs at the isobar corresonding to atmosheric ressure. A small amount of rocessing shrinkage ( s ) occurs on oening the mould and cooling the moulding to ambient temerature. An imortant arameter is the time of delay in comression t determined from the -- grah. he end of comression (oint ) should occur when the mould is sealed, otherwise the lastic may ass through into the runner or further, in the direction of the nozzle. Usually the relations between ressure, secific volume and temerature of the olymers are determined exerimentally. We can also use for this urose mathematical equations (theoretical and exerimental) which are an extension of the thermodynamic equation of state. HEORY OF HEOREICAL EQUAIONS he theoretical equations are based mainly on the equation known from statistical mechanics relating the ressure to secific volume and temerature (ref. ): ln Z = k r () Figure. Diagram of injection-comression moulding rocess: a injection hase; b comression hase (s comression ga comensating for volume contraction) (ref. 9) where k is the Boltzmann constant and Z is a distribution function defined at the atomic level. Direct determination of the distribution function is very difficult, requiring certain assumtions regarding the structure of the macromolecules and intermolecular forces. /78 International Polymer Science and echnology, Vol. 9, No. 7,

4 HE SIMHA-SOMCYNSKY EQUAION Simha and Somcynsky (ref. 6) modified the cellular model roosed by Lennard-Jones and Devonshire (ref. 7). hey defined the magnitude y which is indeendent of temerature, taking the volume of free saces ( holes ) in the system as (- y). Simha and Somcynsky s hole theory made it ossible to define equations describing the reduced variables. he reducing variables, and characterising each olymer were defined as follows (refs. 8 ): he exression c/s is the so-called coefficient of elasticity which characterises the system. he results obtained with different values of this coefficient coincide, hence it is usually assumed that c/s =. P. Zoller (ref. ) determined the reducing arameters of PFE in the liquid state using equations () and (6). he solution of these equations when = allows us to determine the theoretical isobar corresonding to atmosheric ressure as a function of the reduced temerature (Figure ). N A = Ns = M ( z ) se = ck () () ( z ) e = (4) where ' is the volume of a macromolecular segment, N is the number of macromolecules of the olymer, N A is the Avogadro number, M is the molecular weight of a segment of the macromolecule, e is the minimum intersegmental otential energy, s is the number of segments in the macromolecule, z is the coordination number, k is Boltzmann s constant, and c is the number of external degrees of freedom of the macromolecule. he equation of state for the reduced variables takes the form: = 6 yy ( ) [ ( )]. y. 4 y + y... ( ) () According to the Simha-Somcynsky theory y defines the units of the quasi-crystalline network occuied by segments of the macromolecules; hence (-y) is the region of free saces between the segments and thus when s equation () can be written in the form: [ ( )] = s + y ln y c [ ] y ( y ). 49. ( y ) 6 6 yy y y 6 + ( ) (6) Figure. Isobar of = for PFE determined exerimentally in the molten state (solid oints) and suerimosed on the theoretical ( = ) isobar (continuous line) (ref. 7) he exerimental oints of the relationshi log vs. log are dislaced arallel to the axis according to the method of suerimosition, until satisfactory results are obtained. Using this method we can use Figure to determine the reducing arameters: = 796K and = m /kg. he reducing arameter of ressure was determined using the Simha-Somcynsky theory for each value of reduced ressure ISING S EQUAION Most equations of state of olymers use a cellular model that requires searation of the internal and external degrees of freedom. he external degrees of freedom attributed to the chain segment of the olymer are lower than in the case of a low-molecular comound. Since Ising s model does not involve a cellular model, there is no need to searate the internal and external degrees of freedom (ref. ). Ising s equation relating to the reduced arameters of olymers in the liquid state takes the form: ln + + ( )+ r = (7) International Polymer Science and echnology, Vol. 9, No. 7, /79

5 where r is the number of units in the crystalline network occuied by the rth mer. he reducing arameters can be related to the molecular weight by the equation: Ising s equation aroximates to the exerimental data much more closely than the comlex equations based on a cellular model. In addition, it can describe the roerties of fluids with a low molecular weight. R M ρ = (8) r Since the arameter r aears in the reduced equation of state (7), the basic requirement of selfsimilar states is not fulfilled. However, for a olymer in the liquid state the value r, and equation (7) reduces to the form: [ ] = ρ + + ln ( ρ )+ ρ (9) hus olymers in the liquid state with a corresondingly high molecular weight must fulfil the basic requirement of corresonding states. his was confirmed exerimentally by I.C. Sanchez and R.H. Lacombe (ref. ). In the grah (Figure 6) it can be seen that in the case of the ten olymers studied the rincile of corresonding states is fulfilled over a wide range of temerature and ressure. he lines are theoretical isobars calculated from equation (9) on the basis of the reduced ressure and exerimental density measurements corresondingly reduced by the reducing arameters. he maximum error in the density calculations is.%, and the average is.%. Exerimental errors in the determination of the density are in the range..%. PASINE AND WARFIELD S EQUAION A three-arameter equation of state exressed with the use of reduced variables was roosed by D.J. Pastine and R.W. Warfield (ref. ).his was designed for olymers in the liquid state on the basis of the theory of a vibrating chain in a rigid cylinder. It can be reresented by the general relationshi: ( ) = ( )+ ( ), A () where A is a volume function, is the internal ressure, deendent on the internal energy of the olymer. Using the Simha-Somcynsky relationshi under conditions of atmosheric ressure this equation of state can be rewritten in the form: ( ) = ( )+ ln, ln, () In equation () the values (,) later referred to as and reresent the coefficients of this equation. Assuming that f ( ) ( ) = A () and using equations () and () we obtain the relationshi: ( ) ( ), f = ln () in which f() has the ressure dimension. Pastine and Warfield studied olymers in the ressure range MPa, at a temerature exceeding (u to C) the temerature of the corresonding hase change of the given olymer. From exerimental data they determined f() in the form: Figure 6. Grahical reresentation of the rincile of corresonding states for olymers. he lines are theoretical isobars calculated from equation (9) for various values of :.,., (ref. 8). ( ) = f B (4) where B is a arameter showing the ressure magnitude. /8 International Polymer Science and echnology, Vol. 9, No. 7,

6 he above-mentioned authors defined B, and as reducing arameters, determining them by extraolation to a temerature of C and a ressure of zero. Substituting f() into equation of state () we obtain: ln B = () Introducing the reduced variables = / B, = /, = / simlifies equation () to the form: = ln (6) which fulfils the rincile of corresonding states. In order to comare the values obtained from equation (6) with the exerimental data (ref. ), first of all the secific volume and temerature by substituting into equation () =. hen the arameter B was determined, lotting the relationshi = f( / ) for a constant secific volume. he reducing arameters were established by comaring the values of the secific volumes found exerimentally with the values calculated from equation (6). he results for PE-HD in the form of grahs of the theoretical and exerimental isotherms are shown in Figure 7. Corresonding results for the isobars are shown in Figure 8. In both these cases the accuracy of comarison between the exerimental and theoretical data is about. cm /g. Figure 8. HDPE isobars corresonding to ressure (in MPa) : a., b 4, c 8, d, e 6, f (the circles indicate exerimental data; the lines are theoretical, determined using equation (6) (ref. 9) WHIAKER AND GRISKEY S EQUAION his equation was evolved on the basis of the theory of corresonding states (ref. 4). Considering the coefficient of comression: Z = (7) R it was observed that the relationshi between Z and the reduced temerature (/ g ) shows a set of corresonding curves (dislaced arallel to themselves) determined for different olymers under atmosheric ressure (Figure 9). Figure 7. HDPE isotherms corresonding to temerature: a 4. C, b.9 C, c 6.4 C, d 7. C, e 8.9 C, f 89.6 C, g 99.7 C (the circles denote exerimental data, and the continuous line are theoretical, determined using equation (6) (ref. 9) Figure 9.Grah of comressibility coefficient (Z) vs. reduced temerature (/ g ) at atmosheric ressure for different olymers (ref. ) International Polymer Science and echnology, Vol. 9, No. 7, /8

7 From the data in Figure 9 we obtained the following general relationshi between the temerature ( in K), ressure ( in Pa) and secific volume (V in m ): V. =. 94 ρ n g m+ R (8) where ρ is the density at C and ressure. MPa, g is the glass transition temerature, R is the universal gas constant in J/(kg.K), and m and n are equation arameters. he mean error of the data calculated from equation (8) and the exerimental data is %. It is only in the melting region of the olymers that equation (8) does not corresond to the exerimental data there is a discontinuity in the case of all of the studied olymers having a clearly defined melting range. Finally, on differentiating the Whitaker-Griskey equation and taking definite values of the constants and exonents, we get the relationshi: x V K y = (9) g in which ω is a constant value for one kilogram of the olymer (m /kg). Substituting in equation () the exression R M = R () where R' is the individual gas constant exressed in J/ (kg.k), we obtain a thermodynamic equation of state of olymers in the following form: ( + πi )( ω ) = R (4) Equations () and (4) are extremely imortant as regards measurement ossibilities. he Sencer-Gilmore model is of course one of the oldest models, but it still continues to be used in studies of olymers (refs. 7 ). ait s equation ait s equation resents the change in secific volume along the isotherm as a function of two variable arameters B() and C() (refs. 9 and ): where x and y are exonents reresenting ressure functions, K is a function of density at C and at ressure of. MPa (refs. 4).,, Cln ( ) = ( ) + ( ) B () REVIEW OF EXPERIMENAL EQUAIONS he Sencer-Gilmore equation A thermodynamic equation of state of olymers arising from the Van der Waals equation was roosed by Sencer and Gilmore (refs. ) in the form: ( + πi) ( M ω M) = R () where is the external absolute ressure, π i is the internal ressure (cohesion ressure) taken as being indeendent of the volume, M is the secific volume of a single kilomole of olymer under ressure and at temerature, ω M is a coefficient reresenting the constant volume of the olymer macromolecules at a temerature of absolute zero. If we consider the relation between the secific volume for one kilomole, or M = M () where M is the molar mass in g/mol, then equation () takes the form: ( + πi ) ω ( ) = R M () where (, ) is the secific volume at ressure and temerature, C is a universal constant equal to.984, and B() is a arameter defined by the relation ( ) = ( ) B B ex B (6) where B and B are characteristic constants of the given olymer. he volume under atmosheric ressure denoted by (,) and determined by extraolation can be reresented as a olynomial at a temerature < m ( ) = (7), A A A A... or as an exonent, where > m ( ) = ( ) (8), ex α where is the secific volume at atmosheric ressure, α is the coefficient of volume exansion at atmosheric ressure. he authors of ref. 9 have determined the arameter B of several olymers in the liquid and solid states. able shows average values of this arameter in a temerature corresonding to the liquid state of the selected olymers. /8 International Polymer Science and echnology, Vol. 9, No. 7,

8 able. Exerimental and theoretical values of arameter B of olymers in the liquid state (ref. 9) P olymer and corresonding theoretical equation, C PS - B = 4 ex (-4.4 x ) PMMA - B = 8 ex (-6.7 x ) PVC - B = ex (-.6 x ) Exerimental Parameter B, MPa heoretical he maximum deviations between the theoretical calculations in able and the exerimental results for PS, PMMA and PVC are.96%,.% and.8% resectively. CONCLUSION here are many mathematical forms of the thermodynamic equation of state of olymers. he reason for this variety is the various aroaches to the roblem and its solution. Also, an equation of this kind must corresond to the criteria listed below, which are difficult to meet at the same time (ref. 4): - it must be valid over a wide range of ressure and temeratures; - its mathematical form must be generalised so that it can be used for many olymers differing in their roerties and states of aggregation; - it must be relatively simle and easy to use; - it must also enable the calculation of thermodynamic relationshis of the olymer when only hysical roerties that are relatively easy to measure are known, such as the glass transition temerature or melting oint at atmosheric ressure and density at C and under atmosheric ressure. Equations fulfilling all these criteria usually relate to articular grous of olymers either in the solid or liquid state, but it is rarely ossible to describe the behaviour of a articular olymer in both these states. For this reason the existing equations are continually modified and imroved in order to obtain the most universal models. In the field of rocessing use has been made of exerimental equations, mainly Gilmore and Sencer s equation and ait s equation, in which the constants have been emloyed for he conditions of a articular conversion rocess. he relations between these conditions are highly comlex, and for this reason the use of the equations continues to be restricted, and the literature covering these roblems is inadequate. In conversion rocesses it is also imortant to know the derivatives of the secific volume as a function of ressure and temerature. Using the first derivative we can determine the coefficients of volume exansion and the coefficients of comressibility which lay an imortant art in the designing of dimensions of injection mouldings. he exerimental equations of state on differentiation determine the values of these coefficients. Examination of the literature shows among other things the continuously oor knowledge of the relationshi between equations of state and the fundamentals of highmolecular lastics rocessing. In connection with this results will be resented in another article of our own investigations into the effect of cooling of injection mouldings on the thermodynamic equation of state. REFERENCES. R. Sikora Fundamentals of rocessing of highmolecular lastics, Lublin Polytechnic, 99,.8.. J. Szarawara, Alied chemical International Polymer Science and echnology, Vol. 9, No. 7, /8

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