- Email: [email protected]
Universal behavior of linear and star-shaped polymers using relations derived from the blob theory and two parameter theory
European Polymer Journal 37 (2001) 989±993
Universal behavior of linear and star-shaped polymers using relations derived from the blob theory and two parameter theory Dimitrios Papanagopoulos a,b, Anastasios Dondos a,b,* b
a Department of Chemical Engineering, University of Patras, 26500 Patras, Greece Institute of Chemical Engineering and High Temperature Processes, P.O. Box 1414, 26500 Patras, Greece
Received 14 March 2000; received in revised form 15 August 2000; accepted 8 September 2000
Abstract The viscometric expansion factor of linear and star-shaped polymers is related to the number of blobs of their chains. It was found that this relation is universal for these two dierent polymer architectures. Using samples of linear and star-shaped polymers we have also established a second universal law relating the hydrodynamic volume of the chains to their critical overlap concentration c . Ó 2001 Elsevier Science Ltd. All rights reserved.
1. Introduction The description of the variation of the expansion or the hydrodynamic volume of the macromolecular chains using universal parameter, has attracted the interest of many polymer scientists. For these universal relations the two parameters theory has been used [1,2] as well as relations derived from the blob theory [2,3]. Let us indicate here the universal relation between the hydrodynamic volume, using the Fox±Flory equation [4], by Benoit et al. [5,6], and the elution volume in gel permeation chromatography (GPC) for chains of dierent architectures or dierent chemical nature. In most cases, in order to describe the expansion of dierent macromolecular chains by a universal manner, dierent scaled parameters [7±9] or reduced blob parameters [10±15] are used during the last two decades. In this work we shall use reduced blob parameters and a parameter derived from the two parameter theory
* Corresponding author. Department of Chemical Engineering, University of Patras, 26500 Patras, Greece. Tel.: +30-61997-652; fax: +30-61-997-266. E-mail address: [email protected] (A. Dondos).
in order to describe the universal behavior of macromolecular chains diering not only on their chemical nature but also on their architecture i.e. linear and starshaped chains.
2. Experimental part The linear and star-shaped polymers, were prepared using anionic polymerization at ``Charles Sadron'' Institute in Strasbourg. The polydispersity of the samples, as determined by GPC, was always lower than 1.1 for the linear samples and lower than 1.2 for the starshaped polymers [16]. The star-shaped polymers were prepared by using the ``living'' PS branches to initiate the polymerization of 1,4-divinyl benzene, which forms the core of the star polymers (arm-®rst method) [17,18]. The viscosity measurements were conducted with an automated Shott-Gerate viscometer using Ubbelohde type viscometers for serious dilutions. The molecular masses of the samples were measured using static light scattering in tetrahydrofuran, with a RTG Sematech (France) photogoniometer equipped with an He±Ne laser at 628.3 nm.
0014-3057/01/$ - see front matter Ó 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 0 1 4 - 3 0 5 7 ( 0 0 ) 0 0 2 0 5 - 6
990
D. Papanagopoulos, A. Dondos / European Polymer Journal 37 (2001) 989±993
3. Theory and procedure The following equation, proposed by Han [19], 3m 1:5 4 1 m 2 m N a3g 1 2m 1 m 1 Nc relates the viscometric expansion factor of a macromolecular chain, ag , to the number of the blobs, N =Nc , of which this chain consists. In this equation N is the number of the statistical segments of which a given chain consists, Nc is the characteristic number of statistical segments of which one blob of this chain consists and m is the excluded volume parameter. This parameter is related to the exponent of the Mark±Houwink±Sakurada (MHS) equation, a, by the relation 3m 1 a. Using Eq. (1) we can obtain the characteristic number Nc of a given polymer dissolved in a given solvent if for this polymer±solvent system we know the unperturbed dimensions parameter, Kh , or the intrinsic viscosities values, gH , in a theta solvent in order to obtain a3g a3g g=gh g= Kh M 1=2 . For this polymer±solvent system the value of m is obtained applying the MHS equation and from the Stockmayer±Fixman±Burchard (SFB) equation [20,21], gM
1=2
KH 0:51UBM 1=2
2
(where M is the molecular mass) we obtain the value of KH . We must note that in order to apply the MHS and the SFB equations, measurements of intrinsic viscosities must be performed in a given solvent with an homologous series of fractions of a given polymer. In this work we will use not only a homologous series of linear polymers but also of star-shaped polymers: fractions having the same number of branches but diering in their molecular masses. In order to obtain the number of the statistical segments, N, of which one chain consists, necessary for the application of Eq. (1), we divide the molecular mass of this chain by the molecular mass of one statistical segment, ms . The value of ms is obtained multiplying the statistical segment length, A, by the molecular mass per contour length of the chain, ML . The value of A (Kuhn statistical segment) is obtained by the relation 2=3 KH ML 3 A U in which KH is the unperturbed dimensions parameter, as we have already mentioned, and U is the FloryÕs parameter. In the case of linear polymers the value of U is taken equal to 2:5 1023 mol 1 while in the case of starshaped polymers the value of U changes with the number of their branches [22]. As we have already mentioned, the hydrodynamic volume of the macromolecular chains has already been
used [5,6] in order to present by universal manner the calibration of the GPC results. More precisely, the hydrodynamic volume is expressed by gM. In this work we will correlate the hydrodynamic volume with the critical concentration c introduced by De Gennes [3], and separating the dilute regime from that of semi-dilute solutions. Using the Fox±Flory equation [4] g U
r2 3=2 M
4
(where r is the end-to-end distance) the hydrodynamic volume of the macromolecular chain, which is expressed by r2 3=2 can also be expressed by gM=U. A number of works [23±25] has related the critical concentration c , introduced by De Gennes [3], with the dimensions of the polymer chains expressed by their molecular mass or by their segment density. It is evident that the increase of the chain dimensions causes a decrease of c . In this work the critical concentration c will be correlated to the hydrodynamic volume of the chains expressed by gM=U and we will see that this correlation is the same for dierent linear polymers as well as for star-shaped polymers.
4. Results and discussion 4.1. Application of Eq. (1) using linear and star-shaped polymers The viscometric results are taken from the literature and are treated, according to the procedure already described, in order to obtain the characteristic number of statistical segments, Nc , from Eq. (1). The obtained parameters from the viscometric results, for the systems linear PS-tetrahydrofurane [26], 3-arm star PS-toluene [27], 12-arm star PS-toluene [27], 18-arm star poly(butadiene) (PB)-cyclohexane [28] and poly(a-methyl styrene) (PamS)-toluene [29] are given in Table 1. In Fig. 1 (curve A) we present the variation of the viscometric expansion factor, a3g , as a function of the number of blobs, N =Nc , according to Eq. (1) for all the fractions of the above polymers. As we can see in this ®gure all the data points lie on the same straight line the slope of which is equal to 0.21 while the prediction of the slope from Eq. (1) is equal to 0.215 (slope 3m-1:5 1:715±1:5 0:215). In the following we apply Eq. (1) using the viscometric results obtained with values of four other polymer±solvent systems in which the exponents of the MHS equation are almost identical (a 0:745 or 0.75). The four polymer±solvent systems are: linear PS-benzene (region of very high molecular masses [30]), linear polyisoprene (PIP)-toluene [31], 4-arm star PIP-toluene [31] and 6-arm star PIP-toluene [31]. The necessary
D. Papanagopoulos, A. Dondos / European Polymer Journal 37 (2001) 989±993
991
Table 1 Exponent of the MHS equation a, unperturbed dimensions parameter KH , FloryÕs parameter U, statistical segment length A, molecular mass of one statistical segment ms and characteristic number of statistical segments (statistical segment of which one blob consists) Nc , for nine polymer±solvent systems Systems a KH 102 Nc Refs. U 10 23 A (A) ms (ml g 1 )
PS/THF PamS/toluene 3-Arm PS/toluene 12-Arm PS/toluene 18-Arm PB/cyclohex. PS/benzene PIP/toluene 4-Arm PIP/toluene 6-Arm PIP/toluene
0.715 0.710 0.715 0.715 0.715 0.75 0.745 0.745 0.745
8.0 7.8 7.1 3.45 5.6 8.9 13.0 11.0 8.25
(cgs)
2.5 2.5 3.0 7.15 8.2 2.5 2.5 3.5 4.85
19.6 21.6 17.6 5.55 1.8 21.0 11.0 7.5 5.0
823 1014 740 233 20 884 179 122 81.5
3.1 3.0 3.5 18 110 3.6 6.5 12.0 18.0
[26] [29] [27] [27] [28] [30] [31] [31] [31]
Fig. 1. Variation of a3g as a function of the number of blobs, N =Nc , for the systems: curve A; (n): linear PS-THF, (e): 3-arm PStoluene, (+): 12-arm PS-toluene, (s): 18-arm PB-cyclohexane and (h): PamS-toluene. curve B; ( ): linear PS-benzene (very high molecular mass region), (d): linear PIP-toluene, (,): 4-arm PIP-toluene and ( ): 6-arm PIP-toluene.
parameters obtained for these systems in order to apply Eq. (1), are given also in Table 1. As we can see in Fig. 1 (curve B), the points obtained with the linear and the star-shaped polymers lie on the same straight line with slope of 0.254. The MHS equation exponent equal to 0.75 suggests a slope, according to Eq. (1), of 0.25. The experimental points obtained with fractions having a small number of statistical segments, and consequently a low number of blobs, deviate from the straight lines in Fig. 1. In this region of low number of
statistical segments, we are below the onset of the appearance of the power law between g and M. 4.2. Relation between the critical concentration c with the hydrodynamic volume of linear and star-shaped polymers The relation between the critical concentration c , separating the regime of dilute solutions from semidilute solutions, and the segment density of the macromolecular chains, expressed by the inverse of the
992
D. Papanagopoulos, A. Dondos / European Polymer Journal 37 (2001) 989±993
Fig. 2. Variation of the critical concentration c as a function of the molecular mass for linear PS: ( ) and for star-shaped PS: ( ) samples, in ethyl acetate at 25°C.
Fig. 3. Variation of the hydrodynamic volume as a function of c for linear PS: ( ), linear PMMA: (d) and for star-shaped PS: ( ) samples.
D. Papanagopoulos, A. Dondos / European Polymer Journal 37 (2001) 989±993
intrinsic viscosity, 1=g, has already been studied [23,24]. We have also studied [25] the dependence of c with 1=g for dierent linear polymer±solvent systems, and we have found a universal relation between these two parameters. In this work we will study the relation between c and the hydrodynamic volume of macromolecular chains of linear as well as star-shaped polymers. As we have already mentioned, according to the Fox±Flory equation, the hydrodynamic volume can be expressed by the quantity gM=U. The U value will be taken for the linear polymers equal to 2:5 1023 while for the star-shaped polymers U varies with the number of branches and it will be taken from the article of Roovers et al. [22]. More precisely, for the 7-arm, 9-arm, 17-arm and 19-arm PS U is taken equal to 6 1023 , 6:5 1023 , 8 1023 , and 8:3 1023 respectively. In Fig. 2 we present the variation of c as a function of the molecular mass for the system linear PS-ethyl acetate. In the same ®gure we present experimental points obtained with star PS samples dissolved also in ethyl acetate. As it is already observed [25,32] with the linear fractions of PS we obtain a straight line the slope of which is equal to the value of the exponent of the MHS equation with a negative sign. As we can see in this ®gure the points corresponding to the star-shaped PS samples deviate from the straight line obtained with the linear fractions. If we now express the size of the macromolecular chains not by their molecular mass but by their hydrodynamic volume, gM=U, we obtain a universal relation not only for the two dierent linear polymers (PS and PMMA) but also for the star-shaped polymers. This universal behavior is presented in Fig. 3 in which the variation of the hydrodynamic volume of dierent fractions of linear PS and PMMA and of starshaped PS samples is given as a function of the critical concentration c . The presentation of the obtained points in Fig. 3 in semi-logarithmic scale is somehow parallel to the presentation of the GPC [5,6] in which the molecular mass of the hydrodynamic volume gM is presented as a function of the elution volume. In our case on the x-axis we give the critical concentration c while in the GPC results, as we have already mentioned, on the x-axis the elution volume is given. Both these two parameters strongly depend on the hydrodynamic volume of the chains. In conclusion, we have shown that there is a universal behavior of linear and star-shaped polymers when we correlate (a) the viscometric expansion factor of the chains to the number of the blobs of which they consist and (b) the hydrodynamic volume of the chains to the critical overlapping concentration c . In this work, re-
993
lations derived either from the two parameters theory or the blob theory are used and the obtained results show that these two theories do not con¯ict.
References [1] Flory PG. Principles of polymer chemistry. Ithaca, New York: Cornell University Press; 1953. [2] Yamakawa H. Modern theory of polymer solutions. New York: Harper and Row; 1971. [3] De Gennes PG. Scaling concepts in polymer physics. Ithaca, New York: Cornell University Press; 1979. [4] Fox TG, Flory PJ. J Phys Colloid Chem 1949;53:197. [5] Beno^õt H, Grubisic Z, Rempp P, Decker D, Zilliox JG. J Chim Phys 1966;63:1507. [6] Grubisic Z, Rempp P, Beno^õt H. J Polym Sci Part B 1967;5:573. [7] Park IH, Kim J, Chang T. Macromolecules 1992;25:7300. [8] Hayward RC, Graessley WW. Macromolecules 1999;32: 3502. [9] Grassley WW, Hayward RC, Grest GS. Macromolecules 1999;32:3510. [10] Adam M, Delsanti M. Macromolecules 1997;10:1229. [11] Vidakovic P, Rondelez F. Macromolecules 1985;18:700. [12] Daoud M, Jannink G. J Phys Fr 1978;39:331. [13] Akcasu AZ, Han CC. Macromolecules 1979;12:276. [14] Dondos A. J Polym Sci Polym Phys Ed 1990;28:2435. [15] Dondos A. Macromolecules 1993;26:3966. [16] Papanagopoulos D. Thesis, University of Patras, 1995. [17] Zilliox JG. Makromol Chem 1972;156:121. [18] Rein D, Lamps JP, Rempp P, Lutz P, Papanagopoulos D, Tsitsilianis C. Acta Polymer 1993;44:225. [19] Han CC. Polymer 1979;20:1083. [20] Stockmayer WH, Fixman M. J Chem Sci C 1963;1:137. [21] Burchard W. Makromol Chem 1961;50:20. [22] Roovers J, Hadzichristidis N, Fetters LJ. Macromolecules 1983;16:214. [23] Grassley WW. Polymer 1980;21:258. [24] Vidakovic P, Allain C, Rondelez F. Macromolecules 1982;15:1571. [25] Papanagopoulos D, Dondos A. Polymer 1995;36:369. [26] Lapp A, Strazielle C. Makromol Chem 1983;184:1877. [27] Khasat N, Pennisi RW, Hadjichristidis N, Fetters LJ. Macromolecules 1988;21:1100. [28] Roovers J, Martin JE. J Polym Sci Polym Phys Ed 1989;27:2513. [29] Siriani AF, Worsfold DJ, Bywater S. Trans Faraday Soc 1959;55:2124. [30] Einaga Y, Miyiaki Y, Fugita H. J Polym Sci Polym Phys Ed 1979;17:2103. [31] Hadjichristidis N, Roovers JEL. J Polym Sci Polym Phys Ed 1974;12:2521. [32] Nystr om B, Roots J. Prog Polym Sci 1982;8:333.
Universal behavior of linear and star-shaped polymers using relations derived from the blob theory and two parameter theory Dimitrios Papanagopoulos a,b, Anastasios Dondos a,b,* b
a Department of Chemical Engineering, University of Patras, 26500 Patras, Greece Institute of Chemical Engineering and High Temperature Processes, P.O. Box 1414, 26500 Patras, Greece
Received 14 March 2000; received in revised form 15 August 2000; accepted 8 September 2000
Abstract The viscometric expansion factor of linear and star-shaped polymers is related to the number of blobs of their chains. It was found that this relation is universal for these two dierent polymer architectures. Using samples of linear and star-shaped polymers we have also established a second universal law relating the hydrodynamic volume of the chains to their critical overlap concentration c . Ó 2001 Elsevier Science Ltd. All rights reserved.
1. Introduction The description of the variation of the expansion or the hydrodynamic volume of the macromolecular chains using universal parameter, has attracted the interest of many polymer scientists. For these universal relations the two parameters theory has been used [1,2] as well as relations derived from the blob theory [2,3]. Let us indicate here the universal relation between the hydrodynamic volume, using the Fox±Flory equation [4], by Benoit et al. [5,6], and the elution volume in gel permeation chromatography (GPC) for chains of dierent architectures or dierent chemical nature. In most cases, in order to describe the expansion of dierent macromolecular chains by a universal manner, dierent scaled parameters [7±9] or reduced blob parameters [10±15] are used during the last two decades. In this work we shall use reduced blob parameters and a parameter derived from the two parameter theory
* Corresponding author. Department of Chemical Engineering, University of Patras, 26500 Patras, Greece. Tel.: +30-61997-652; fax: +30-61-997-266. E-mail address: [email protected] (A. Dondos).
in order to describe the universal behavior of macromolecular chains diering not only on their chemical nature but also on their architecture i.e. linear and starshaped chains.
2. Experimental part The linear and star-shaped polymers, were prepared using anionic polymerization at ``Charles Sadron'' Institute in Strasbourg. The polydispersity of the samples, as determined by GPC, was always lower than 1.1 for the linear samples and lower than 1.2 for the starshaped polymers [16]. The star-shaped polymers were prepared by using the ``living'' PS branches to initiate the polymerization of 1,4-divinyl benzene, which forms the core of the star polymers (arm-®rst method) [17,18]. The viscosity measurements were conducted with an automated Shott-Gerate viscometer using Ubbelohde type viscometers for serious dilutions. The molecular masses of the samples were measured using static light scattering in tetrahydrofuran, with a RTG Sematech (France) photogoniometer equipped with an He±Ne laser at 628.3 nm.
0014-3057/01/$ - see front matter Ó 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 0 1 4 - 3 0 5 7 ( 0 0 ) 0 0 2 0 5 - 6
990
D. Papanagopoulos, A. Dondos / European Polymer Journal 37 (2001) 989±993
3. Theory and procedure The following equation, proposed by Han [19], 3m 1:5 4 1 m 2 m N a3g 1 2m 1 m 1 Nc relates the viscometric expansion factor of a macromolecular chain, ag , to the number of the blobs, N =Nc , of which this chain consists. In this equation N is the number of the statistical segments of which a given chain consists, Nc is the characteristic number of statistical segments of which one blob of this chain consists and m is the excluded volume parameter. This parameter is related to the exponent of the Mark±Houwink±Sakurada (MHS) equation, a, by the relation 3m 1 a. Using Eq. (1) we can obtain the characteristic number Nc of a given polymer dissolved in a given solvent if for this polymer±solvent system we know the unperturbed dimensions parameter, Kh , or the intrinsic viscosities values, gH , in a theta solvent in order to obtain a3g a3g g=gh g= Kh M 1=2 . For this polymer±solvent system the value of m is obtained applying the MHS equation and from the Stockmayer±Fixman±Burchard (SFB) equation [20,21], gM
1=2
KH 0:51UBM 1=2
2
(where M is the molecular mass) we obtain the value of KH . We must note that in order to apply the MHS and the SFB equations, measurements of intrinsic viscosities must be performed in a given solvent with an homologous series of fractions of a given polymer. In this work we will use not only a homologous series of linear polymers but also of star-shaped polymers: fractions having the same number of branches but diering in their molecular masses. In order to obtain the number of the statistical segments, N, of which one chain consists, necessary for the application of Eq. (1), we divide the molecular mass of this chain by the molecular mass of one statistical segment, ms . The value of ms is obtained multiplying the statistical segment length, A, by the molecular mass per contour length of the chain, ML . The value of A (Kuhn statistical segment) is obtained by the relation 2=3 KH ML 3 A U in which KH is the unperturbed dimensions parameter, as we have already mentioned, and U is the FloryÕs parameter. In the case of linear polymers the value of U is taken equal to 2:5 1023 mol 1 while in the case of starshaped polymers the value of U changes with the number of their branches [22]. As we have already mentioned, the hydrodynamic volume of the macromolecular chains has already been
used [5,6] in order to present by universal manner the calibration of the GPC results. More precisely, the hydrodynamic volume is expressed by gM. In this work we will correlate the hydrodynamic volume with the critical concentration c introduced by De Gennes [3], and separating the dilute regime from that of semi-dilute solutions. Using the Fox±Flory equation [4] g U
r2 3=2 M
4
(where r is the end-to-end distance) the hydrodynamic volume of the macromolecular chain, which is expressed by r2 3=2 can also be expressed by gM=U. A number of works [23±25] has related the critical concentration c , introduced by De Gennes [3], with the dimensions of the polymer chains expressed by their molecular mass or by their segment density. It is evident that the increase of the chain dimensions causes a decrease of c . In this work the critical concentration c will be correlated to the hydrodynamic volume of the chains expressed by gM=U and we will see that this correlation is the same for dierent linear polymers as well as for star-shaped polymers.
4. Results and discussion 4.1. Application of Eq. (1) using linear and star-shaped polymers The viscometric results are taken from the literature and are treated, according to the procedure already described, in order to obtain the characteristic number of statistical segments, Nc , from Eq. (1). The obtained parameters from the viscometric results, for the systems linear PS-tetrahydrofurane [26], 3-arm star PS-toluene [27], 12-arm star PS-toluene [27], 18-arm star poly(butadiene) (PB)-cyclohexane [28] and poly(a-methyl styrene) (PamS)-toluene [29] are given in Table 1. In Fig. 1 (curve A) we present the variation of the viscometric expansion factor, a3g , as a function of the number of blobs, N =Nc , according to Eq. (1) for all the fractions of the above polymers. As we can see in this ®gure all the data points lie on the same straight line the slope of which is equal to 0.21 while the prediction of the slope from Eq. (1) is equal to 0.215 (slope 3m-1:5 1:715±1:5 0:215). In the following we apply Eq. (1) using the viscometric results obtained with values of four other polymer±solvent systems in which the exponents of the MHS equation are almost identical (a 0:745 or 0.75). The four polymer±solvent systems are: linear PS-benzene (region of very high molecular masses [30]), linear polyisoprene (PIP)-toluene [31], 4-arm star PIP-toluene [31] and 6-arm star PIP-toluene [31]. The necessary
D. Papanagopoulos, A. Dondos / European Polymer Journal 37 (2001) 989±993
991
Table 1 Exponent of the MHS equation a, unperturbed dimensions parameter KH , FloryÕs parameter U, statistical segment length A, molecular mass of one statistical segment ms and characteristic number of statistical segments (statistical segment of which one blob consists) Nc , for nine polymer±solvent systems Systems a KH 102 Nc Refs. U 10 23 A (A) ms (ml g 1 )
PS/THF PamS/toluene 3-Arm PS/toluene 12-Arm PS/toluene 18-Arm PB/cyclohex. PS/benzene PIP/toluene 4-Arm PIP/toluene 6-Arm PIP/toluene
0.715 0.710 0.715 0.715 0.715 0.75 0.745 0.745 0.745
8.0 7.8 7.1 3.45 5.6 8.9 13.0 11.0 8.25
(cgs)
2.5 2.5 3.0 7.15 8.2 2.5 2.5 3.5 4.85
19.6 21.6 17.6 5.55 1.8 21.0 11.0 7.5 5.0
823 1014 740 233 20 884 179 122 81.5
3.1 3.0 3.5 18 110 3.6 6.5 12.0 18.0
[26] [29] [27] [27] [28] [30] [31] [31] [31]
Fig. 1. Variation of a3g as a function of the number of blobs, N =Nc , for the systems: curve A; (n): linear PS-THF, (e): 3-arm PStoluene, (+): 12-arm PS-toluene, (s): 18-arm PB-cyclohexane and (h): PamS-toluene. curve B; ( ): linear PS-benzene (very high molecular mass region), (d): linear PIP-toluene, (,): 4-arm PIP-toluene and ( ): 6-arm PIP-toluene.
parameters obtained for these systems in order to apply Eq. (1), are given also in Table 1. As we can see in Fig. 1 (curve B), the points obtained with the linear and the star-shaped polymers lie on the same straight line with slope of 0.254. The MHS equation exponent equal to 0.75 suggests a slope, according to Eq. (1), of 0.25. The experimental points obtained with fractions having a small number of statistical segments, and consequently a low number of blobs, deviate from the straight lines in Fig. 1. In this region of low number of
statistical segments, we are below the onset of the appearance of the power law between g and M. 4.2. Relation between the critical concentration c with the hydrodynamic volume of linear and star-shaped polymers The relation between the critical concentration c , separating the regime of dilute solutions from semidilute solutions, and the segment density of the macromolecular chains, expressed by the inverse of the
992
D. Papanagopoulos, A. Dondos / European Polymer Journal 37 (2001) 989±993
Fig. 2. Variation of the critical concentration c as a function of the molecular mass for linear PS: ( ) and for star-shaped PS: ( ) samples, in ethyl acetate at 25°C.
Fig. 3. Variation of the hydrodynamic volume as a function of c for linear PS: ( ), linear PMMA: (d) and for star-shaped PS: ( ) samples.
D. Papanagopoulos, A. Dondos / European Polymer Journal 37 (2001) 989±993
intrinsic viscosity, 1=g, has already been studied [23,24]. We have also studied [25] the dependence of c with 1=g for dierent linear polymer±solvent systems, and we have found a universal relation between these two parameters. In this work we will study the relation between c and the hydrodynamic volume of macromolecular chains of linear as well as star-shaped polymers. As we have already mentioned, according to the Fox±Flory equation, the hydrodynamic volume can be expressed by the quantity gM=U. The U value will be taken for the linear polymers equal to 2:5 1023 while for the star-shaped polymers U varies with the number of branches and it will be taken from the article of Roovers et al. [22]. More precisely, for the 7-arm, 9-arm, 17-arm and 19-arm PS U is taken equal to 6 1023 , 6:5 1023 , 8 1023 , and 8:3 1023 respectively. In Fig. 2 we present the variation of c as a function of the molecular mass for the system linear PS-ethyl acetate. In the same ®gure we present experimental points obtained with star PS samples dissolved also in ethyl acetate. As it is already observed [25,32] with the linear fractions of PS we obtain a straight line the slope of which is equal to the value of the exponent of the MHS equation with a negative sign. As we can see in this ®gure the points corresponding to the star-shaped PS samples deviate from the straight line obtained with the linear fractions. If we now express the size of the macromolecular chains not by their molecular mass but by their hydrodynamic volume, gM=U, we obtain a universal relation not only for the two dierent linear polymers (PS and PMMA) but also for the star-shaped polymers. This universal behavior is presented in Fig. 3 in which the variation of the hydrodynamic volume of dierent fractions of linear PS and PMMA and of starshaped PS samples is given as a function of the critical concentration c . The presentation of the obtained points in Fig. 3 in semi-logarithmic scale is somehow parallel to the presentation of the GPC [5,6] in which the molecular mass of the hydrodynamic volume gM is presented as a function of the elution volume. In our case on the x-axis we give the critical concentration c while in the GPC results, as we have already mentioned, on the x-axis the elution volume is given. Both these two parameters strongly depend on the hydrodynamic volume of the chains. In conclusion, we have shown that there is a universal behavior of linear and star-shaped polymers when we correlate (a) the viscometric expansion factor of the chains to the number of the blobs of which they consist and (b) the hydrodynamic volume of the chains to the critical overlapping concentration c . In this work, re-
993
lations derived either from the two parameters theory or the blob theory are used and the obtained results show that these two theories do not con¯ict.
References [1] Flory PG. Principles of polymer chemistry. Ithaca, New York: Cornell University Press; 1953. [2] Yamakawa H. Modern theory of polymer solutions. New York: Harper and Row; 1971. [3] De Gennes PG. Scaling concepts in polymer physics. Ithaca, New York: Cornell University Press; 1979. [4] Fox TG, Flory PJ. J Phys Colloid Chem 1949;53:197. [5] Beno^õt H, Grubisic Z, Rempp P, Decker D, Zilliox JG. J Chim Phys 1966;63:1507. [6] Grubisic Z, Rempp P, Beno^õt H. J Polym Sci Part B 1967;5:573. [7] Park IH, Kim J, Chang T. Macromolecules 1992;25:7300. [8] Hayward RC, Graessley WW. Macromolecules 1999;32: 3502. [9] Grassley WW, Hayward RC, Grest GS. Macromolecules 1999;32:3510. [10] Adam M, Delsanti M. Macromolecules 1997;10:1229. [11] Vidakovic P, Rondelez F. Macromolecules 1985;18:700. [12] Daoud M, Jannink G. J Phys Fr 1978;39:331. [13] Akcasu AZ, Han CC. Macromolecules 1979;12:276. [14] Dondos A. J Polym Sci Polym Phys Ed 1990;28:2435. [15] Dondos A. Macromolecules 1993;26:3966. [16] Papanagopoulos D. Thesis, University of Patras, 1995. [17] Zilliox JG. Makromol Chem 1972;156:121. [18] Rein D, Lamps JP, Rempp P, Lutz P, Papanagopoulos D, Tsitsilianis C. Acta Polymer 1993;44:225. [19] Han CC. Polymer 1979;20:1083. [20] Stockmayer WH, Fixman M. J Chem Sci C 1963;1:137. [21] Burchard W. Makromol Chem 1961;50:20. [22] Roovers J, Hadzichristidis N, Fetters LJ. Macromolecules 1983;16:214. [23] Grassley WW. Polymer 1980;21:258. [24] Vidakovic P, Allain C, Rondelez F. Macromolecules 1982;15:1571. [25] Papanagopoulos D, Dondos A. Polymer 1995;36:369. [26] Lapp A, Strazielle C. Makromol Chem 1983;184:1877. [27] Khasat N, Pennisi RW, Hadjichristidis N, Fetters LJ. Macromolecules 1988;21:1100. [28] Roovers J, Martin JE. J Polym Sci Polym Phys Ed 1989;27:2513. [29] Siriani AF, Worsfold DJ, Bywater S. Trans Faraday Soc 1959;55:2124. [30] Einaga Y, Miyiaki Y, Fugita H. J Polym Sci Polym Phys Ed 1979;17:2103. [31] Hadjichristidis N, Roovers JEL. J Polym Sci Polym Phys Ed 1974;12:2521. [32] Nystr om B, Roots J. Prog Polym Sci 1982;8:333.