Analysis of equations of state for polymers

Padilha Júnior, Erlí José; Soares, Rafael de Pelegrini; Cardozo, Nilo Sérgio Medeiros

doi:10.1590/0104-1428.1621

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Seção Técnica • Polímeros 25 (3) • May-Jun 2015 • https://doi.org/10.1590/0104-1428.1621 copy

Analysis of equations of state for polymers

Authorship SCIMAGO INSTITUTIONS RANKINGS

Abstract

In the literature there are several studies comparing the accuracy of various models in describing the PvT behavior of polymers. However, most of these studies do not provide information about the quality of the estimated parameters or the sensitivity of the prediction of thermodynamic properties to the parameters of the equations. Furthermore, there are few studies exploring the prediction of thermal expansion and compression coefficients. Based on these observations, the objective of this study is to deepen the analysis of Tait, HH (Hartmann-Haque), MCM (modified cell model) and SHT (simplified hole theory) equations of state in predicting the PvT behavior of polymers, for both molten and solid states. The results showed that all equations of state provide an adequate description of the PvT behavior in the molten state, with low standard deviations in the estimation of parameters, adequate sensitivity of their parameters and plausible prediction of specific volume, thermal expansion and isothermal compression coefficients. In the solid state the Tait equation exhibited similar performance to the molten state, while HH showed satisfactory results for amorphous polymers and difficulty in adjusting the PvT curve for semicrystalline polymers.

Keywords:
equation of state; PvT behavior; polymer

Polymer	Amorphous						Semicrystalline
Polymer	PS	PC	PMMA	PCHMA	PnBMA	PoMS	iPP	LPE	PEO	PA6	PLA	PBS-B
Reference	28	29	30	30	30	31	29	32	7	21	32	32
Pressure range (MPa)	0.1-200	0.1-200	0.1-180	0.1-180	0.1-180	0.1-180	0.1-200	0.1-200	0.1-200	0.1-190	0.1-200	0.1-200
Temperature range (K)	280-468	313-603	353-423	357-453	285-355	302-470	313-573	313-493	313-493	364-586	313-493	313-493
Initial temperature of analysis (K)	468	313	353	357	285	302	313	313	313	364	313	313
Initial pressure of analysis (MPa)	0.1	0.1	0.1	0.1	0.1	0.1	0.1	0.1	0.1	0.1	0.1	0.1
Number of experimental points in solid state used for parameter estimation	33	26	36	30	29	25	28	41	27	35	56	16
Number of experimental points in molten state used for parameter estimation	28	30	20	26	22	26	25	29	43	17	21	20
Number of experimental points in solid state used for the analysis of the prediction accuracy	33	24	-	30	-	-	28	-	-	35	-	15
Number of experimental points in molten state used for the analysis of the prediction accuracy	28	32	-	26	-	-	24	-	-	17	-	19
Experimental data variance ( $σ_{e x p}^{2}$ )	0.0003	0.001	0.0004	0.0004	0.0004	0.0004	0.001	0.002	0.002	0.0003	0.002	0.002

EoS	Parameter	PS		PC		PMMA		PCHMA		PnBMA		PoMS		Mean
EoS	Parameter	$a_{i}$	$σ_{a_{i}}$ (%)	$a_{i}$	$σ_{a_{i}}$ (%)	$a_{i}$	$σ_{a_{i}}$ (%)	$a_{i}$	$σ_{a_{i}}$ (%)	$a_{i}$	$σ_{a_{i}}$ (%)	$a_{i}$	$σ_{a_{i}}$ (%)	$σ_{a_{i}}$ (%)
Tait	$b_{1 m}$ (cm³/g)	0.9767	0.0015	0.8590	0.0025	0.8603	0.0024	0.9318	0.0017	0.9469	0.0014	1.0046	0.0013	0.0018
	$b_{2 m}$ (cm³/gK)	0.000506	0.0461	0.000553	0.0392	0.000511	0.1042	0.000596	0.0477	0.000628	0.0532	0.000547	0.0492	0.0566
	$b_{3 m}$ (MPa)	154.65	0.0672	151.39	0.1062	277.56	0.1512	161.98	0.0629	192.89	0.0998	184.57	0.0882	0.0959
	$b_{4 m}$ (1/K)	0.0030	0.2749	0.0034	0.2218	0.0075	0.6696	0.0052	0.1845	0.0047	0.4290	0.0062	0.2388	0.3364
	$b_{1 s}$ (cm³/g)	0.9748	0.0015	0.8575	0.0035	0.8625	0.0007	0.9303	0.0014	0.9479	0.0011	1.0045	0.0018	0.0017
	$b_{2 s}$ (cm³/gK)	0.000213	0.1116	0.000192	0.2435	0.000275	0.1665	0.000248	0.1118	0.000376	0.3084	0.000232	0.0996	0.1736
	$b_{3 s}$ (MPa)	275.87	0.0645	249.21	0.1005	257.75	0.0303	255.61	0.0486	223.53	0.0373	276.20	0.0649	0.0577
	$b_{4 s}$ (1/K)	0.0018	0.6080	0.0021	0.7384	0.0045	0.3877	0.0038	0.3082	0.0036	0.8858	0.00067	1.2603	0.6981
	$b_{5}$ (K)	370.86	0.0026	417.06	0.0059	375.30	0.0034	383.84	0.0030	296.66	0.0039	401.21	0.0028	0.0036
	$b_{6}$ (K/MPa)	0.3924	0.0211	0.2687	0.0885	0.3485	0.0692	0.2665	0.0441	0.2225	0.0528	0.3361	0.0350	0.0518
HH	$B_{0_{m}}$ (MPa)	2953.3	1.3157	3470.2	2.6453	5238.2	5.0184	3115.0	1.3573	3654.1	2.5917	3154.5	2.1292	2.5096
	$v_{0_{m}}$ (cm³/g)	0.8753	0.6226	0.7413	1.3425	0.7570	0.7592	0.8192	0.7058	0.8431	0.6388	0.8917	0.6503	0.7865
	$T_{0_{m}}$ (K)	1602.1	0.2884	1471.8	0.6255	1464.9	0.7126	1471.8	0.3327	1256.7	0.4040	1631.2	0.3724	0.4559
	$B_{0_{s}}$ (MPa)	4278.5	3.6067	3858.2	7.8331	3867.7	3.2444	4172.8	3.7264	3499.0	3.3230	4251.8	4.6353	4.3948
	$v_{0_{s}}$ (cm³/g)	0.9218	0.5102	0.8107	1.4375	0.8158	0.5568	0.8785	0.6056	0.8891	0.7577	0.9399	0.6413	0.7515
	$T_{0_{s}}$ (K)	2552.0	0.4872	2914.7	1.3898	2592.6	0.4136	2656.6	0.5196	1863.6	0.4846	2453.5	0.7396	0.6724
MCM	$P^{}$ (MPa)*	532.04	0.4534	707.66	0.9018	1017.5	2.1312	594.83	0.8393	656.45	0.6733	616.17	0.9196	0.9864
	$v^{*}$ (cm³/g)	0.8690	0.0822	0.7357	0.1447	0.7463	0.2813	0.8101	0.1353	0.8392	0.1177	0.8745	0.1612	0.1537
	$T^{}$ (K)*	6933.9	0.3985	6441.5	0.5280	6178.8	1.1607	6298.7	0.5389	5427.0	0.5621	6704.6	0.6911	0.6466
SHT	$P^{}$ (MPa)*	420.57	1.0068	675.78	2.1852	887.13	6.4028	518.74	2.1551	506.85	1.3689	577.23	4.3201	2.9065
	$v^{*}$ (cm³/g)	0.9242	0.1769	0.7620	0.3835	0.7798	0.8736	0.8491	0.3381	0.8937	0.2329	0.9032	0.7175	0.4538
	$T^{}$ (K)*	4009.2	0.8431	3353.9	1.3560	3311.9	3.4399	3425.4	1.3222	3168.3	1.1022	3429.9	2.7175	1.7968

EoS	Parameter	iPP		LPE		PEO		PA6		PLA		PBS-B		Mean
EoS	Parameter	$a_{i}$	$σ_{a_{i}}$ (%)	$a_{i}$	$σ_{a_{i}}$ (%)	$a_{i}$	$σ_{a_{i}}$ (%)	$a_{i}$	$σ_{a_{i}}$ (%)	$a_{i}$	$σ_{a_{i}}$ (%)	$a_{i}$	$σ_{a_{i}}$ (%)	$σ_{a_{i}}$ (%)
Tait	$b_{1 m}$ (cm³/g)	1.3082	0.0021	1.2532	0.0022	1.2301	0.0018	1.0002	0.0030	0.8875	0.0043	0.8837	0.0035	0.0028
	$b_{2 m}$ (cm³/gK)	0.0010	0.0378	0.000957	0.0509	0.000932	0.0304	0.000659	0.0873	0.000712	0.1455	0.000622	0.0788	0.0718
	$b_{3 m}$ (MPa)	66.84	0.0438	93.68	0.0769	113.12	0.0577	132.00	0.0860	100.68	0.1041	152.55	0.1278	0.0827
	$b_{4 m}$ (1/K)	0.0048	0.1116	0.0046	0.2521	0.0044	0.1411	0.0029	0.5355	0.0047	0.5810	0.0040	0.4383	0.3433
	$b_{1 s}$ (cm³/g)	1.1804	0.0033	1.0669	0.0030	1.1666	0.0975	0.9600	0.0020	0.8553	0.0025	0.8361	0.0109	0.0199
	$b_{2 s}$ (cm³/gK)	0.000517	0.0743	0.000453	0.1063	0.000780	1.1831	0.000489	0.0396	0.000372	0.0695	0.000525	0.2850	0.2930
	$b_{3 s}$ (MPa)	110.82	0.0765	228.35	0.0825	173.63	1.3530	125.96	0.0496	148.00	0.0902	198.59	0.3480	0.3333
	$b_{4 s}$ (1/K)	0.0064	0.1556	0.0023	0.5972	0.0029	3.0337	0.0078	0.0830	0.0064	0.1984	0.0057	1.1048	0.8621
	$b_{5}$ (K)	452.86	0.0055	405.47	0.0061	365.09	0.0068	501.95	0.0020	440.44	0.0056	385.88	0.0064	0.0054
	$b_{6}$ (K/MPa)	0.0057	4.1917	0.2107	0.1129	0.2043	0.1164	0.0835	0.1127	0.0048	4.9477	0.1254	0.1897	1.6119
	$b_{7}$ (cm³/g)	0.3644	0.2083	0.4848	0.1569	0.0565	1.9711	0.0406	0.0709	0.0327	0.1187	0.0133	2.7994	0.8876
	$b_{8}$ (1/K)	0.1429	0.1600	0.2893	0.1153	0.0210	1.4194	0.0890	0.0718	0.4684	0.2048	0.1785	1.4186	0.5650
	$b_{9}$ (1/MPa)	0.1133	0.2824	0.0762	0.1072	0.0074	0.5776	0.0029	0.2204	0.0081	0.2998	0.0111	1.3604	0.4746
HH	$B_{0_{m}}$ (MPa)	1877.2	0.9060	2514.9	1.5167	2581.2	1.3639	4330.2	2.6157	3235.6	1.6826	3648.7	3.1454	1.8717
	$v_{0_{m}}$ (cm³/g)	1.0864	1.5161	1.0496	1.4684	1.0605	1.1865	0.8137	1.5780	0.7196	1.8242	0.7573	1.8053	1.5631
	$T_{0_{m}}$ (K)	1370.7	0.3162	1273.9	0.4684	1277.8	0.4502	1437.2	0.7388	1242.7	0.6546	1327.6	0.8649	0.5822
	$B_{0_{s}}$ (MPa)	2689.0	3.2847	5508.9	5.8402	5242.2	2.0723	5634.4	1.7457	4112.4	4.1033	6393.9	9.2255	4.3786
	$v_{0_{s}}$ (cm³/g)	1.0241	1.4413	0.9064	1.0336	0.9097	1.6756	0.7828	0.5857	0.7426	1.0285	0.6945	2.0437	1.3014
	$T_{0_{s}}$ (K)	1574.8	0.9417	1284.6	0.9761	830.57	0.5881	1396.3	0.3584	1576.0	0.9510	1170.2	1.4734	0.8815
MCM	$P^{}$ (MPa)*	406.30	1.4660	497.88	1.7795	516.67	0.9554	882.19	1.3540	668.30	3.7141	717.11	1.6693	1.8231
	$v^{*}$ (cm³/g)	1.0656	0.1957	1.0435	0.2740	1.0492	0.1417	0.8110	0.1867	0.7141	0.5219	0.7510	0.2474	0.2612
	$T^{}$ (K)*	5825.1	0.5207	5601.7	0.8363	5521.4	0.4740	6392.2	0.4969	5457.8	1.3499	5774.1	0.8529	0.7551
SHT	$P^{}$ (MPa)*	445.72	3.5869	509.45	4.7145	509.45	2.4290	922.41	3.4977	714.20	9.5570	678.21	4.2874	4.6788
	$v^{*}$ (cm³/g)	1.0880	0.5217	1.0730	0.7374	1.0836	0.3770	0.8285	0.4958	0.7288	1.3734	0.7783	0.6555	0.6935
	$T^{*}$ (K)	2929.4	1.3930	2864.6	2.2278	2855.9	1.2422	3209.4	1.3034	2737.2	3.5220	3016.0	2.2253	1.9856

Polymer	Molten													Solid
	EoS												$F_{c}$ ^a a Degrees of freedom used: 28, 32, 26, 24, 17 and 19 for PS, PC, PCHMA, iPP, PA6 and PBS-B, respectively; bDegrees of freedom used: 33, 24, 30, 28, 35 and 15 for PS, PC, PCHMA, iPP, PA6 and PBS-B, respectively.	EoS						$F_{c}$ b
	Tait			HH			MCM			SHT				Tait			HH
	MRD (%)	$R^{2}$	$F_{0}$	MRD (%)	$R^{2}$	$F_{0}$	MRD (%)	$R^{2}$	$F_{0}$	MRD (%)	$R^{2}$	$F_{0}$		MRD (%)	$R^{2}$	$F_{0}$	MRD (%)	$R^{2}$	$F_{0}$
PS	0.0394	0.9997	8.38×10^–4	0.0581	0.9992	1.81×10^–9	0.0323	0.9997	6.38×10^–4	0.0520	0.9995	0.0013	1.4792	0.0201	0.9998	1.78×10^–4	0.0294	0.9994	4.22×10^–4	1.4393
PC	0.0723	0.9994	4.84×10^–4	0.1077	0.9990	7.27×10^–4	0.1333	0.9976	0.0018	0.1141	0.9988	9.08×10^–4	1.4465	0.0557	0.9982	1.97×10^–4	0.1395	0.9877	0.0012	1.5200
PCHMA	0.0364	0.9998	6.11×10^–4	0.1136	0.9984	4.03×10^–9	0.0808	0.9992	0.0020	0.0612	0.9995	0.0014	1.4984	0.0639	0.9976	0.0020	0.1085	0.9937	0.0049	1.4620
IPP	0.1028	0.9996	0.0017	0.1600	0.9988	0.0043	0.1320	0.9991	0.0031	0.1478	0.9990	0.0035	1.5200	0.4300	0.9249	0.0933	0.9429	0.8659	0.1532	1.4792
PA6	0.1070	0.9978	0.0129	0.1010	0.9980	0.0101	0.0804	0.9992	0.0040	0.1059	0.9987	0.0067	1.6253	0.1964	0.9920	0.0297	0.6487	0.9316	0.2388	1.4259
PBS-B	0.0444	0.9998	1.66×10^–4	0.0583	0.9996	2.48×10^–4	0.0612	0.9995	2.75×10^–4	0.0733	0.9994	3.40×10^–4	1.5891	0.0890	0.9978	5.40×10^–4	0.1741	0.9899	0.0021	1.6689
Amorphous Mean	-	-	-	-	-	-	-	-	-	-	-	-	-	0.0466	0.9985	-	0.0925	0.9936	-	-
Semicrystalline Mean	-	-	-	-	-	-	-	-	-	-	-	-	-	0.2385	0.9716	-	0.5886	0.9291	-	-
Mean	0.0671	0.9994	-	0.0998	0.9988	-	0.0867	0.9991	-	0.0924	0.9992	-	-	-	-	-	-	-	-	-

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