- Email: [email protected]
Density function analysis of single polymer chain
Chaos, Solitons and Fractals 32 (2007) 1250–1257 www.elsevier.com/locate/chaos
Density function analysis of single polymer chain Hisataka Kato, Miki Wadati
*
Department of Physics, Graduate School of Science, University of Tokyo, Bunkyo-ku, Hongo 7-3-1, Tokyo 113-0033, Japan Accepted 29 November 2005
Communicated by Prof. M.S. El Naschie
Abstract We study the conformational properties of single polymer chain both with and without electric charge. We propose a new method by considering the conformational properties of single polymer chain. In stead of traditional approaches, we introduce the density function of polymer segments. And we calculate the physical quantities from this new approach. For the most simple and the most important case, we confirm that Flory’s exponents m reproduce 3/5, which is completely same as the Flory’s. We discuss the problems related to the improvement of this new concept that will be useful in future studies. Ó 2005 Elsevier Ltd. All rights reserved.
1. Introduction Traditionally, single polymer chain has been treated by beads-spring model and its extensions [1–3]. While it is easy to evaluate the conformational entropy in these models, it is difficult to evaluate the conformational energy because of non-Markovian process in the random-walk pictures. Recently, Cannavacciuolo and Pedersen [4,5] introduced an empirical expression for the angular correlation function of charged (Debye–Hu¨ckel) wormlike chains (WLC) with excluded-volume (EV) interactions, and analyzed the moments hR2ki by use of the end-to-end distance distribution function f(r) of charged WLC with EV. Their method suggests us the possibility of analyzing the single polymer chain from the viewpoint of segments-density-distribution function: if moments hR2ki of single polymer chain are known, we may know the density distribution of segments. In this paper, we consider a method to treat the single polymer chain with density function q(r), and then express the conformational energy E and entropy S as the functional of the density function q(r). By minimizing the free energy F = E TS or F F =ðk B T Þ, where kB denotes the Boltzmann’s constant, we determine the density function q(r) and the Flory’s exponent m. Since the total segment number N is constant, the density function q(r) is subject to the following constraint: Z qðrÞdr ¼ N . ð1Þ
*
Corresponding author. Tel.: +81 3 5841 7633; fax: +81 3 5841 7587. E-mail address: [email protected] (M. Wadati).
0960-0779/$ - see front matter Ó 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2005.11.084
H. Kato, M. Wadati / Chaos, Solitons and Fractals 32 (2007) 1250–1257
1251
2. Energy and entropy If only two-body interaction u(r1, r2) is concerned, the conformational energy E is described as follows: Z Z 1 E¼ qðr1 Þqðr2 Þuðr1 ; r2 Þdr1 dr2 . 2
ð2Þ
Therefore, for the d-function EV case, uðr1 ; r2 Þ ¼ k B TV c dðr1 r2 Þ;
ð3Þ
we have, EEV ¼
k B TV c 2
Z Z
qðr1 Þqðr2 Þdðr1 r2 Þdr1 dr2 ¼
k B TV c 2
Z
qðrÞ2 dr;
ð4Þ
where Vc denotes the strength of the two-body interaction, that is, the excluded volume. The interaction energy (4) can be modified to include an attractive potential (see, Section 4.2). And for the Screened-Coulomb case, we have ejjr1 r2 j uðr1 ; r2 Þ ¼ a2 k B TlB ; jr1 r2 j Z Z a2 k B TlB ejjr1 r2 j qðr1 Þqðr2 Þ dr1 dr2 ; ESC ¼ 2 jr1 r2 j
ð5Þ ð6Þ
where a is the degree of ionization, and lB = e2/(4pr0kBT) is the Bjerrum length. Now we consider the conformational entropy as the functional of segment-density q(r). There is no a priori choice of this entropy, so we first refer Flory’s expression, S ¼ k B
3R2 ; 2Nb2
ð7Þ
where R is the end-to-end distance, and b is the segment length. For the case of the ideal chain, the end-to-end distance R is proportional to the gyration radius rg [3], N D E 1 X ðRn RG Þ2 ; ð8Þ N n¼1 P where RG ¼ N1 Nn¼1 Rn is the center of the mass of the polymer. In the density-picture, we can assume that RG = 0. Therefore, the gyration radius can be written as Z 1 qðrÞr2 dr. r2g ¼ ð9Þ N
r2g ¼
Inserting this expression to Eq. (7), we can express the conformational entropy in the form of the functional of density q(r) as follows: Z kB S¼ 2 2 qðrÞr2 dr. ð10Þ bN Although this expression has not been derived exactly, it has some interesting properties. First, it is similar to the famous von Neumann’s entropy, S N ¼ Trq ln q.
ð11Þ
If we replace the q in the log-term by the Flory’s expression, we obtain the similar expression of the entropy (10). Second, we relate it to the elastic property. It is well known that single polymer chain has the entropy elasticity, and instead of the entropy, we can consider the elastic energy of spring (beads-spring model). From this viewpoint, the r2-entropy is plausible. For these reasons, we adopt the expression (10) in what follows. Because of spherical symmetry of the entropy, the density function q(r) also has the spherical symmetry q(r) = q(r). The Flory exponent m, R / Nm, is determined from the following relation: Z 1 qðrÞr2 dr / N 2m . ð12Þ r2g ¼ N
1252
H. Kato, M. Wadati / Chaos, Solitons and Fractals 32 (2007) 1250–1257
3. Single polymer chain with excluded-volume interaction We assume that the two-body interaction is repulsive, that is, V c > 0. In this section, the conformational energy is Z k B TV c qðrÞ2 dr EEV ¼ 2 and the entropy is Z kB qðrÞr2 dr. S¼ 2 2 bN We minimize the ‘‘free energy’’ F ¼ ðE TSÞ=ðk B T Þ with the normalization condition, Z qðrÞdr ¼ N .
ð13Þ
ð14Þ
ð15Þ
ð16Þ
3.1. Density function of segments We employ the variation method with the Lagrange multiplier l to obtain dðF lN Þ r2 ¼ V c qðrÞ þ 2 2 l ¼ 0; dqðrÞ bN 1 r2 l 2 2 . qðrÞ ¼ Vc bN
ð17Þ ð18Þ
It is interesting that this equation also appears in the analysis of the Bose–Einstein condensation (see Appendix A). Since the density is positive definite, some restrictions are attached to the range of r: pffiffiffi 1 0 6 r 6 R ¼ min bN l; bN . ð19Þ 2 In the second condition, R ¼ 12 bN means the rod-like limit, since the total length L = bN corresponds to the diameter of the distribution in this limit. But for a while, this condition is disregarded. From the normalization condition, we have the expression of the Lagrange multiplier l: Z R 4pR3 l R2 N¼ ð20Þ qðrÞ4pr2 dr ¼ 2 2 ; V c 3 5b N 0 3R2 3V c N l¼ 2 2þ . ð21Þ 4pR3 5b N Substituting (21) for (19), we get 3R2 3V c b2 N 3 þ ; 5 4pR3 1=5 15 2=5 3=5 V 1=5 N . R2 ¼ c b 8p
R2 ¼
ð22Þ ð23Þ
Now we take account of the condition for the total-length-constant into consideration. From Eqs. (19) and (23), we have 1=5 15 1 2=5 3=5 V 1=5 N 6 bN . ð24Þ R¼ c b 8p 2 This condition is satisfied for the large N. If the two-body interaction is so strong that it holds V c > 601 pb3 N 2 . This indicates that R outgrows the stretching limit 12 bN . In this case, we obtain 1 pffiffiffi 1 ð25Þ R ¼ min bN l; bN ¼ bN ; 2 2 that is, the polymer exhibits the behavior like a rod.
H. Kato, M. Wadati / Chaos, Solitons and Fractals 32 (2007) 1250–1257
1253
3.2. Flory exponent For the ordinary cases, since the expression of the existence limit R is, 1=5 15 2=5 3=5 V 1=5 N R¼ c b 8p the Flory exponent m seems to be m = 3/5. In fact, 2=5 3 15 4=5 6=5 r2g hr2 i ¼ V 2=5 N c b 7 8p
ð26Þ
ð27Þ
that is, rg ¼
1=2 1=5 3 15 2=5 3=5 V 1=5 N 0:65 R. c b 7 8p
ð28Þ
Furthermore, we can show that the sth moment hrsi is given by hrs i ¼
2=5 3=5 s 23s=5 151þs=5 ps=5 ðV 1=5 N Þ c b OðN ms Þ. 2 15 þ 8s þ s
ð29Þ
Therefore, for any order of moments, we obtain 3 m¼ . 5
ð30Þ
3.3. Comparison with Flory’s theory In the simple Flory’s theory [1,2], the energy and entropy are respectively, EFlory 6=5 1=5 EV ¼ V 2=5 N ; c b kB T S Flory 3 ¼ V 2=5 b6=5 N 1=5 ; 2 c kB
ð31Þ ð32Þ
while in our theory those are given by Z EEV V c 152=5 EFlory 6=5 1=5 EV V 2=5 b N ¼ 0:234 ¼ qðrÞ2 dr ¼ 1=5 ; c kB T 2 kBT 72 p2=5 S 1 ¼ 2 2 kB bN
Z
r2 qðrÞdr ¼
3152=5 S Flory V 2=5 b6=5 N 1=5 ¼ 0:234 . 1=5 2=5 c kB 142 p
ð33Þ
ð34Þ
As is well known, the Flory’s theory overestimates both the conformational energy and the conformational entropy. From the comparison, we see that both quantities are fairly improved. 3.4. Dimensionality d = 2 Let us now turn to the other important case of dimensionality d = 2. For this case, the variational calculation is the same to the case of dimensionality d = 3 (Eqs. (18) and (19)). From these equations, the normalization condition is written as Z R0 pR02 R02 N¼ ð35Þ qðrÞ2prdr ¼ 2l 2 2 ; 2V c bN 0 R02 V cN ð36Þ l ¼ 2 2 þ 02 . pR 2b N
1254
H. Kato, M. Wadati / Chaos, Solitons and Fractals 32 (2007) 1250–1257
Substituting (36) into (19), we get R02 V c b2 N 3 ; þ R02 ¼ 2 pR2 1=4 2 R0 ¼ . V c b2 N 3 p The conformational energy and the conformational entropy are given as follows: rffiffiffiffiffiffiffiffiffiffiffiffi Z EFlory EEV;2d V c 2V c N EV;2d 2 ¼ q dr ¼ ; ¼ 0:217 kBT 2 kB T 9pb2 ffiffiffiffiffiffiffiffiffiffiffi ffi r Z S Flory S EV;2d 1 2V c N EV;2d r2 qðrÞdr ¼ ¼ 2 2 ; ¼ 0:217 2 kB kB bN 9pb
ð37Þ ð38Þ
ð39Þ ð40Þ
where we have used the Flory’s expression to compare with our results. Actually, the Flory’s theory in two-dimension gives rffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffi 1=4 EFlory 2 3V c N S Flory 3V c N EV;2d 2 3 2d V cb N R¼ ¼ ; ¼ ; . ð41Þ 3 kBT kB 2b2 2b2 Again we observe the improvements of the conformational energy and the conformational entropy. Finally, we calculate the swelling exponent m, Z 1 23þs=4 ps=4 ðV c b2 N 3 Þs=4 rs qðrÞdr ¼ . hrs i ¼ N 8 þ 6s þ s2
ð42Þ
We find that m = 3/4 for the case of dimensionality d = 2. This result is also common to the Flory’s theory m = 3/(d + 2). In particular, we have r2g hr2 i ¼
R02 . 3
ð43Þ
4. Some generalizations In this section, we generalize the discussion into more general cases where the interaction depends only on the distance of two segments, uðr1 ; r2 Þ ¼ k B TU ðjr1 r2 jÞ.
ð44Þ
The form of U(r) is rather arbitrary but decreases fast enough to be Fourier transformable. By using the variational calculation, we have Z dðF lN Þ r2 ¼ qðr0 ÞU ðjr r0 jÞdr0 þ 2 2 l ¼ 0. ð45Þ dqðrÞ bN We adopt the following definition of the Fourier-transformation: Z f^ ðqÞ ¼ f ðrÞeiqr dr; Z 1 f^ ðqÞeiqr dq. f ðrÞ ¼ ð2pÞ3
ð46Þ ð47Þ
Then we apply this transformation to the integral Eq. (45). However, the term r2 in the integral equation cannot be transformed directly, because the r-integral is not convergent. To avoid this difficulty, we introduce a trick: we consider the d-function, Z eiqr dr ¼ 2pdðqÞ ð48Þ and operate the Laplacian $2 in the momentum space on both sides, Z r2 eiqr dr ¼ 2pd00 ðqÞ.
ð49Þ
H. Kato, M. Wadati / Chaos, Solitons and Fractals 32 (2007) 1250–1257
From these relations we formally obtain 00 b ðqÞ ¼ 2p ldðqÞ þ d ðqÞ ; ^ðqÞ U q b2 N 2 00 2p d ðqÞ ^ðqÞ ¼ ldðqÞ þ 2 2 ; q b ðqÞ bN U
1255
ð50Þ ð51Þ
b ðqÞ is the Fourier-transformed potential. We remark that since U is a function of jr1 r2j, the Fourier-transwhere U b ðqÞ depends only on the length of the vector q = jqj. formed potential energy function U Using the above relations, we obtain 2 3 Z b ð0Þ 2 1 U 1 d00 ðqÞ iqr 1 4 r2 5 ldðqÞ þ 2 2 e dq ¼ lþ 2 2 r qðrÞ ¼ ð52Þ 2 2 . b ðqÞ b ð0Þ b ðqÞ bN bN bN U U U q¼0
When we compare this distribution function (52) with the d-function excluded-volume interaction case (18), we find the generalized ‘‘excluded-volume’’ Vfc , Z b ð0Þ ¼ U ðrÞdr. Vfc ¼ U ð53Þ
4.1. Screened-coulomb interaction In this case, Eq. (45) has a form, 4p 2p ^ðqÞ ¼ 2pldðqÞ þ 2 2 d00 ðqÞ; q j2 þ q2 bN j2 þ q2 1 00 ^ðqÞ ¼ ldðqÞ þ d ðqÞ . q 2a2 lB b2 N 2 a2 lB
From these relations, we have Z 1 1 00 1 6 r2 2 2 iqr 2 ðj . þ q Þ ldðqÞ þ d ðqÞ e dq ¼ j l þ qðrÞ ¼ 4pa2 lB 4pa2 lB b2 N 2 b2 N 2 b2 N 2 The normalization condition becomes Z 2R3 j2 R5 j2 R3 l N ¼ qðrÞdr ¼ 2 2 2 2 2 2 þ 2 3a lB a lB b N 5a lB b N
ð54Þ ð55Þ
ð56Þ
ð57Þ
and then the chemical potential is 3R2 6 3a2 lB N 2 2 2þ 2 3 . 2 2 jR 5b N jbN
ð58Þ
We consider again the existence limit: rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 6 3R2 3a2 lB b2 N 3 þ ¼ R; b2 N 2 l þ 2 ¼ 5 j j2 R3 1=5 15 2=5 3=5 2=5 a2=5 l1=5 N b . R¼ B j 2
ð59Þ
l¼
From these equations we obtain 2=5 15 6 4=5 4=5 6=5 l¼ a4=5 l2=5 N b 2 2 2. B j 2 jbN And the second criterion of the existence limit R < bN/2 is written as follows: pffiffiffiffiffi 4 15al1=2 B j> . b3=2 N In this region we also observe m = 3/5.
ð60Þ
ð61Þ
ð62Þ
1256
H. Kato, M. Wadati / Chaos, Solitons and Fractals 32 (2007) 1250–1257
4.2. d-Function EV with slight modification We may include the effects of attractive interaction. We introduce the additional attractive force in addition to the dfunction excluded-volume interaction. The potential, for example, is expressed as uðrÞ ¼ k B TV c dðrÞ k B T ehðr rÞ; where h(x) is the step function, i.e., 1 for x > 0; hðxÞ ¼ 0 for x < 0:
ð63Þ
ð64Þ
In this case, we have b ðqÞ ¼ V c e 4pðqr cosðqrÞ þ sinðqrÞÞ . U q3
ð65Þ
Therefore, we obtain 1 1 ¼ ; b ð0Þ V c 4pr3 e=3 U 1 4pr5 e . r2 ¼ b 5ðV c 4pr3 e=3Þ2 U ðqÞ q¼0 Substituting (66) and (67) into (52), we arrive at 1 1 4pr5 e r2 l . qðrÞ ¼ V c 4pr3 e=3 b2 N 2 5ðV c 4pr3 e=3Þ b2 N 2
ð66Þ ð67Þ
ð68Þ
If we define the following quantities: Vfc ¼ V c 4pr3 e=3;
ð69Þ
1 4pr5 e ~ ¼l 2 2 ; l b N 5ðV c 4pr3 e=3Þ
ð70Þ
we see that this result is similar to the case of the simple EV.
5. Conclusions and discussions In this paper, we have developed a new method which treats single polymer chain by use of the density function. We have applied this approach to some types of interactions, that is, the d-function excluded-volume interaction, the screened-Coulomb interaction and the more general cases. In the traditional pictures, such as freely-jointed-chain model and wormlike chain model, connections of segments are considered first. So these models enable us to include the entropic properties easily, while the energetic properties are difficult to take into account, since the correlation along the polymer is no longer long-ranged in the real space (solvent). This difficulty is often referred to as a non-Markovian process. To avoid this difficulty, we have introduced the density function. In this picture, we can take the energetic properties into account in a simple manner. But this model does not give an intuitive guess of the entropy. Since we like to represent the connected properties in the expression of the entropy, we have suggested the r2-entropy and discussed its justification. By using this picture, we have found the Flory exponent m = 3/5 unless the polymer is overstretching, that is, R < 12 bN . For the case of d-function excluded-volume interaction, this result m = 3/5 is completely same as the Flory’s. But we have found that both the conformational energy and the conformational entropy are fairly improved, since we have considered the fluctuation in the distribution of segments. For the case of screened-Coulomb interaction, we have confirmed that our result R2 N6/5/j4/5 supports the most of simulation results, which come from the linear prediction [6]. From these results, we conclude that our suggestion of r2-entropy is quite appropriate, at least as far as the chain possesses a certain amount of flexibility. And in this regime, the swelling exponent is found to be m = 3/5.
H. Kato, M. Wadati / Chaos, Solitons and Fractals 32 (2007) 1250–1257
1257
Acknowledgements One of the authors (H.K.) would like to thank Dr. J. Ieda and Mr. M. Uchiyama for many useful discussions.
Appendix A. Bose–Einstein condensation within Thomas–Fermi approximation The Bose–Einstein condensate under the isotropic magnetic trap is described by the Gross–Pitaevskii equation, h2 2 1 ð71Þ $ þ mx2 x2 þ U 0 jWðxÞj2 WðxÞ ¼ lWðxÞ. 2 2m This equation is known also as the (static) non-linear Schro¨dinger equation (NLSE). If we neglect the kinetic energy term (Thomas–Fermi approximation), we obtain 1 1 qðxÞ ¼ jWðxÞj2 ¼ ð72Þ l mx2 x2 . U0 2 This equation is the same as (18).
References [1] [2] [3] [4]
Flory PJ. Principles of polymer chemistry. Ithaca (NY): Cornell University Press; 1953. Flory PJ. Statistical mechanics of chain molecules. New York: InterScience; 1977. de Gennes PG. Scaling concepts in polymer physics. Ithaca, NY: Cornell University Press; 1979. Cannavacciuolo L, Pedersen JS. Properties of polyelectrolyte chains from analysis of angular correlation functions. J Chem Phys 2002;117:8973–82. [5] Cannavacciuolo L, Pedersen JS. Moments and distribution function of polyelectrolyte chains. J Chem Phys 2004;120:8862–5. [6] Ullner M. Comments on the scaling behavior of flexible polyelectrolytes within the Debye–Hu¨ckel approximation. J Phys Chem B 2003;107:8097–110. and references therein.
Density function analysis of single polymer chain Hisataka Kato, Miki Wadati
*
Department of Physics, Graduate School of Science, University of Tokyo, Bunkyo-ku, Hongo 7-3-1, Tokyo 113-0033, Japan Accepted 29 November 2005
Communicated by Prof. M.S. El Naschie
Abstract We study the conformational properties of single polymer chain both with and without electric charge. We propose a new method by considering the conformational properties of single polymer chain. In stead of traditional approaches, we introduce the density function of polymer segments. And we calculate the physical quantities from this new approach. For the most simple and the most important case, we confirm that Flory’s exponents m reproduce 3/5, which is completely same as the Flory’s. We discuss the problems related to the improvement of this new concept that will be useful in future studies. Ó 2005 Elsevier Ltd. All rights reserved.
1. Introduction Traditionally, single polymer chain has been treated by beads-spring model and its extensions [1–3]. While it is easy to evaluate the conformational entropy in these models, it is difficult to evaluate the conformational energy because of non-Markovian process in the random-walk pictures. Recently, Cannavacciuolo and Pedersen [4,5] introduced an empirical expression for the angular correlation function of charged (Debye–Hu¨ckel) wormlike chains (WLC) with excluded-volume (EV) interactions, and analyzed the moments hR2ki by use of the end-to-end distance distribution function f(r) of charged WLC with EV. Their method suggests us the possibility of analyzing the single polymer chain from the viewpoint of segments-density-distribution function: if moments hR2ki of single polymer chain are known, we may know the density distribution of segments. In this paper, we consider a method to treat the single polymer chain with density function q(r), and then express the conformational energy E and entropy S as the functional of the density function q(r). By minimizing the free energy F = E TS or F F =ðk B T Þ, where kB denotes the Boltzmann’s constant, we determine the density function q(r) and the Flory’s exponent m. Since the total segment number N is constant, the density function q(r) is subject to the following constraint: Z qðrÞdr ¼ N . ð1Þ
*
Corresponding author. Tel.: +81 3 5841 7633; fax: +81 3 5841 7587. E-mail address: [email protected] (M. Wadati).
0960-0779/$ - see front matter Ó 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2005.11.084
H. Kato, M. Wadati / Chaos, Solitons and Fractals 32 (2007) 1250–1257
1251
2. Energy and entropy If only two-body interaction u(r1, r2) is concerned, the conformational energy E is described as follows: Z Z 1 E¼ qðr1 Þqðr2 Þuðr1 ; r2 Þdr1 dr2 . 2
ð2Þ
Therefore, for the d-function EV case, uðr1 ; r2 Þ ¼ k B TV c dðr1 r2 Þ;
ð3Þ
we have, EEV ¼
k B TV c 2
Z Z
qðr1 Þqðr2 Þdðr1 r2 Þdr1 dr2 ¼
k B TV c 2
Z
qðrÞ2 dr;
ð4Þ
where Vc denotes the strength of the two-body interaction, that is, the excluded volume. The interaction energy (4) can be modified to include an attractive potential (see, Section 4.2). And for the Screened-Coulomb case, we have ejjr1 r2 j uðr1 ; r2 Þ ¼ a2 k B TlB ; jr1 r2 j Z Z a2 k B TlB ejjr1 r2 j qðr1 Þqðr2 Þ dr1 dr2 ; ESC ¼ 2 jr1 r2 j
ð5Þ ð6Þ
where a is the degree of ionization, and lB = e2/(4pr0kBT) is the Bjerrum length. Now we consider the conformational entropy as the functional of segment-density q(r). There is no a priori choice of this entropy, so we first refer Flory’s expression, S ¼ k B
3R2 ; 2Nb2
ð7Þ
where R is the end-to-end distance, and b is the segment length. For the case of the ideal chain, the end-to-end distance R is proportional to the gyration radius rg [3], N D E 1 X ðRn RG Þ2 ; ð8Þ N n¼1 P where RG ¼ N1 Nn¼1 Rn is the center of the mass of the polymer. In the density-picture, we can assume that RG = 0. Therefore, the gyration radius can be written as Z 1 qðrÞr2 dr. r2g ¼ ð9Þ N
r2g ¼
Inserting this expression to Eq. (7), we can express the conformational entropy in the form of the functional of density q(r) as follows: Z kB S¼ 2 2 qðrÞr2 dr. ð10Þ bN Although this expression has not been derived exactly, it has some interesting properties. First, it is similar to the famous von Neumann’s entropy, S N ¼ Trq ln q.
ð11Þ
If we replace the q in the log-term by the Flory’s expression, we obtain the similar expression of the entropy (10). Second, we relate it to the elastic property. It is well known that single polymer chain has the entropy elasticity, and instead of the entropy, we can consider the elastic energy of spring (beads-spring model). From this viewpoint, the r2-entropy is plausible. For these reasons, we adopt the expression (10) in what follows. Because of spherical symmetry of the entropy, the density function q(r) also has the spherical symmetry q(r) = q(r). The Flory exponent m, R / Nm, is determined from the following relation: Z 1 qðrÞr2 dr / N 2m . ð12Þ r2g ¼ N
1252
H. Kato, M. Wadati / Chaos, Solitons and Fractals 32 (2007) 1250–1257
3. Single polymer chain with excluded-volume interaction We assume that the two-body interaction is repulsive, that is, V c > 0. In this section, the conformational energy is Z k B TV c qðrÞ2 dr EEV ¼ 2 and the entropy is Z kB qðrÞr2 dr. S¼ 2 2 bN We minimize the ‘‘free energy’’ F ¼ ðE TSÞ=ðk B T Þ with the normalization condition, Z qðrÞdr ¼ N .
ð13Þ
ð14Þ
ð15Þ
ð16Þ
3.1. Density function of segments We employ the variation method with the Lagrange multiplier l to obtain dðF lN Þ r2 ¼ V c qðrÞ þ 2 2 l ¼ 0; dqðrÞ bN 1 r2 l 2 2 . qðrÞ ¼ Vc bN
ð17Þ ð18Þ
It is interesting that this equation also appears in the analysis of the Bose–Einstein condensation (see Appendix A). Since the density is positive definite, some restrictions are attached to the range of r: pffiffiffi 1 0 6 r 6 R ¼ min bN l; bN . ð19Þ 2 In the second condition, R ¼ 12 bN means the rod-like limit, since the total length L = bN corresponds to the diameter of the distribution in this limit. But for a while, this condition is disregarded. From the normalization condition, we have the expression of the Lagrange multiplier l: Z R 4pR3 l R2 N¼ ð20Þ qðrÞ4pr2 dr ¼ 2 2 ; V c 3 5b N 0 3R2 3V c N l¼ 2 2þ . ð21Þ 4pR3 5b N Substituting (21) for (19), we get 3R2 3V c b2 N 3 þ ; 5 4pR3 1=5 15 2=5 3=5 V 1=5 N . R2 ¼ c b 8p
R2 ¼
ð22Þ ð23Þ
Now we take account of the condition for the total-length-constant into consideration. From Eqs. (19) and (23), we have 1=5 15 1 2=5 3=5 V 1=5 N 6 bN . ð24Þ R¼ c b 8p 2 This condition is satisfied for the large N. If the two-body interaction is so strong that it holds V c > 601 pb3 N 2 . This indicates that R outgrows the stretching limit 12 bN . In this case, we obtain 1 pffiffiffi 1 ð25Þ R ¼ min bN l; bN ¼ bN ; 2 2 that is, the polymer exhibits the behavior like a rod.
H. Kato, M. Wadati / Chaos, Solitons and Fractals 32 (2007) 1250–1257
1253
3.2. Flory exponent For the ordinary cases, since the expression of the existence limit R is, 1=5 15 2=5 3=5 V 1=5 N R¼ c b 8p the Flory exponent m seems to be m = 3/5. In fact, 2=5 3 15 4=5 6=5 r2g hr2 i ¼ V 2=5 N c b 7 8p
ð26Þ
ð27Þ
that is, rg ¼
1=2 1=5 3 15 2=5 3=5 V 1=5 N 0:65 R. c b 7 8p
ð28Þ
Furthermore, we can show that the sth moment hrsi is given by hrs i ¼
2=5 3=5 s 23s=5 151þs=5 ps=5 ðV 1=5 N Þ c b OðN ms Þ. 2 15 þ 8s þ s
ð29Þ
Therefore, for any order of moments, we obtain 3 m¼ . 5
ð30Þ
3.3. Comparison with Flory’s theory In the simple Flory’s theory [1,2], the energy and entropy are respectively, EFlory 6=5 1=5 EV ¼ V 2=5 N ; c b kB T S Flory 3 ¼ V 2=5 b6=5 N 1=5 ; 2 c kB
ð31Þ ð32Þ
while in our theory those are given by Z EEV V c 152=5 EFlory 6=5 1=5 EV V 2=5 b N ¼ 0:234 ¼ qðrÞ2 dr ¼ 1=5 ; c kB T 2 kBT 72 p2=5 S 1 ¼ 2 2 kB bN
Z
r2 qðrÞdr ¼
3152=5 S Flory V 2=5 b6=5 N 1=5 ¼ 0:234 . 1=5 2=5 c kB 142 p
ð33Þ
ð34Þ
As is well known, the Flory’s theory overestimates both the conformational energy and the conformational entropy. From the comparison, we see that both quantities are fairly improved. 3.4. Dimensionality d = 2 Let us now turn to the other important case of dimensionality d = 2. For this case, the variational calculation is the same to the case of dimensionality d = 3 (Eqs. (18) and (19)). From these equations, the normalization condition is written as Z R0 pR02 R02 N¼ ð35Þ qðrÞ2prdr ¼ 2l 2 2 ; 2V c bN 0 R02 V cN ð36Þ l ¼ 2 2 þ 02 . pR 2b N
1254
H. Kato, M. Wadati / Chaos, Solitons and Fractals 32 (2007) 1250–1257
Substituting (36) into (19), we get R02 V c b2 N 3 ; þ R02 ¼ 2 pR2 1=4 2 R0 ¼ . V c b2 N 3 p The conformational energy and the conformational entropy are given as follows: rffiffiffiffiffiffiffiffiffiffiffiffi Z EFlory EEV;2d V c 2V c N EV;2d 2 ¼ q dr ¼ ; ¼ 0:217 kBT 2 kB T 9pb2 ffiffiffiffiffiffiffiffiffiffiffi ffi r Z S Flory S EV;2d 1 2V c N EV;2d r2 qðrÞdr ¼ ¼ 2 2 ; ¼ 0:217 2 kB kB bN 9pb
ð37Þ ð38Þ
ð39Þ ð40Þ
where we have used the Flory’s expression to compare with our results. Actually, the Flory’s theory in two-dimension gives rffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffi 1=4 EFlory 2 3V c N S Flory 3V c N EV;2d 2 3 2d V cb N R¼ ¼ ; ¼ ; . ð41Þ 3 kBT kB 2b2 2b2 Again we observe the improvements of the conformational energy and the conformational entropy. Finally, we calculate the swelling exponent m, Z 1 23þs=4 ps=4 ðV c b2 N 3 Þs=4 rs qðrÞdr ¼ . hrs i ¼ N 8 þ 6s þ s2
ð42Þ
We find that m = 3/4 for the case of dimensionality d = 2. This result is also common to the Flory’s theory m = 3/(d + 2). In particular, we have r2g hr2 i ¼
R02 . 3
ð43Þ
4. Some generalizations In this section, we generalize the discussion into more general cases where the interaction depends only on the distance of two segments, uðr1 ; r2 Þ ¼ k B TU ðjr1 r2 jÞ.
ð44Þ
The form of U(r) is rather arbitrary but decreases fast enough to be Fourier transformable. By using the variational calculation, we have Z dðF lN Þ r2 ¼ qðr0 ÞU ðjr r0 jÞdr0 þ 2 2 l ¼ 0. ð45Þ dqðrÞ bN We adopt the following definition of the Fourier-transformation: Z f^ ðqÞ ¼ f ðrÞeiqr dr; Z 1 f^ ðqÞeiqr dq. f ðrÞ ¼ ð2pÞ3
ð46Þ ð47Þ
Then we apply this transformation to the integral Eq. (45). However, the term r2 in the integral equation cannot be transformed directly, because the r-integral is not convergent. To avoid this difficulty, we introduce a trick: we consider the d-function, Z eiqr dr ¼ 2pdðqÞ ð48Þ and operate the Laplacian $2 in the momentum space on both sides, Z r2 eiqr dr ¼ 2pd00 ðqÞ.
ð49Þ
H. Kato, M. Wadati / Chaos, Solitons and Fractals 32 (2007) 1250–1257
From these relations we formally obtain 00 b ðqÞ ¼ 2p ldðqÞ þ d ðqÞ ; ^ðqÞ U q b2 N 2 00 2p d ðqÞ ^ðqÞ ¼ ldðqÞ þ 2 2 ; q b ðqÞ bN U
1255
ð50Þ ð51Þ
b ðqÞ is the Fourier-transformed potential. We remark that since U is a function of jr1 r2j, the Fourier-transwhere U b ðqÞ depends only on the length of the vector q = jqj. formed potential energy function U Using the above relations, we obtain 2 3 Z b ð0Þ 2 1 U 1 d00 ðqÞ iqr 1 4 r2 5 ldðqÞ þ 2 2 e dq ¼ lþ 2 2 r qðrÞ ¼ ð52Þ 2 2 . b ðqÞ b ð0Þ b ðqÞ bN bN bN U U U q¼0
When we compare this distribution function (52) with the d-function excluded-volume interaction case (18), we find the generalized ‘‘excluded-volume’’ Vfc , Z b ð0Þ ¼ U ðrÞdr. Vfc ¼ U ð53Þ
4.1. Screened-coulomb interaction In this case, Eq. (45) has a form, 4p 2p ^ðqÞ ¼ 2pldðqÞ þ 2 2 d00 ðqÞ; q j2 þ q2 bN j2 þ q2 1 00 ^ðqÞ ¼ ldðqÞ þ d ðqÞ . q 2a2 lB b2 N 2 a2 lB
From these relations, we have Z 1 1 00 1 6 r2 2 2 iqr 2 ðj . þ q Þ ldðqÞ þ d ðqÞ e dq ¼ j l þ qðrÞ ¼ 4pa2 lB 4pa2 lB b2 N 2 b2 N 2 b2 N 2 The normalization condition becomes Z 2R3 j2 R5 j2 R3 l N ¼ qðrÞdr ¼ 2 2 2 2 2 2 þ 2 3a lB a lB b N 5a lB b N
ð54Þ ð55Þ
ð56Þ
ð57Þ
and then the chemical potential is 3R2 6 3a2 lB N 2 2 2þ 2 3 . 2 2 jR 5b N jbN
ð58Þ
We consider again the existence limit: rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 6 3R2 3a2 lB b2 N 3 þ ¼ R; b2 N 2 l þ 2 ¼ 5 j j2 R3 1=5 15 2=5 3=5 2=5 a2=5 l1=5 N b . R¼ B j 2
ð59Þ
l¼
From these equations we obtain 2=5 15 6 4=5 4=5 6=5 l¼ a4=5 l2=5 N b 2 2 2. B j 2 jbN And the second criterion of the existence limit R < bN/2 is written as follows: pffiffiffiffiffi 4 15al1=2 B j> . b3=2 N In this region we also observe m = 3/5.
ð60Þ
ð61Þ
ð62Þ
1256
H. Kato, M. Wadati / Chaos, Solitons and Fractals 32 (2007) 1250–1257
4.2. d-Function EV with slight modification We may include the effects of attractive interaction. We introduce the additional attractive force in addition to the dfunction excluded-volume interaction. The potential, for example, is expressed as uðrÞ ¼ k B TV c dðrÞ k B T ehðr rÞ; where h(x) is the step function, i.e., 1 for x > 0; hðxÞ ¼ 0 for x < 0:
ð63Þ
ð64Þ
In this case, we have b ðqÞ ¼ V c e 4pðqr cosðqrÞ þ sinðqrÞÞ . U q3
ð65Þ
Therefore, we obtain 1 1 ¼ ; b ð0Þ V c 4pr3 e=3 U 1 4pr5 e . r2 ¼ b 5ðV c 4pr3 e=3Þ2 U ðqÞ q¼0 Substituting (66) and (67) into (52), we arrive at 1 1 4pr5 e r2 l . qðrÞ ¼ V c 4pr3 e=3 b2 N 2 5ðV c 4pr3 e=3Þ b2 N 2
ð66Þ ð67Þ
ð68Þ
If we define the following quantities: Vfc ¼ V c 4pr3 e=3;
ð69Þ
1 4pr5 e ~ ¼l 2 2 ; l b N 5ðV c 4pr3 e=3Þ
ð70Þ
we see that this result is similar to the case of the simple EV.
5. Conclusions and discussions In this paper, we have developed a new method which treats single polymer chain by use of the density function. We have applied this approach to some types of interactions, that is, the d-function excluded-volume interaction, the screened-Coulomb interaction and the more general cases. In the traditional pictures, such as freely-jointed-chain model and wormlike chain model, connections of segments are considered first. So these models enable us to include the entropic properties easily, while the energetic properties are difficult to take into account, since the correlation along the polymer is no longer long-ranged in the real space (solvent). This difficulty is often referred to as a non-Markovian process. To avoid this difficulty, we have introduced the density function. In this picture, we can take the energetic properties into account in a simple manner. But this model does not give an intuitive guess of the entropy. Since we like to represent the connected properties in the expression of the entropy, we have suggested the r2-entropy and discussed its justification. By using this picture, we have found the Flory exponent m = 3/5 unless the polymer is overstretching, that is, R < 12 bN . For the case of d-function excluded-volume interaction, this result m = 3/5 is completely same as the Flory’s. But we have found that both the conformational energy and the conformational entropy are fairly improved, since we have considered the fluctuation in the distribution of segments. For the case of screened-Coulomb interaction, we have confirmed that our result R2 N6/5/j4/5 supports the most of simulation results, which come from the linear prediction [6]. From these results, we conclude that our suggestion of r2-entropy is quite appropriate, at least as far as the chain possesses a certain amount of flexibility. And in this regime, the swelling exponent is found to be m = 3/5.
H. Kato, M. Wadati / Chaos, Solitons and Fractals 32 (2007) 1250–1257
1257
Acknowledgements One of the authors (H.K.) would like to thank Dr. J. Ieda and Mr. M. Uchiyama for many useful discussions.
Appendix A. Bose–Einstein condensation within Thomas–Fermi approximation The Bose–Einstein condensate under the isotropic magnetic trap is described by the Gross–Pitaevskii equation, h2 2 1 ð71Þ $ þ mx2 x2 þ U 0 jWðxÞj2 WðxÞ ¼ lWðxÞ. 2 2m This equation is known also as the (static) non-linear Schro¨dinger equation (NLSE). If we neglect the kinetic energy term (Thomas–Fermi approximation), we obtain 1 1 qðxÞ ¼ jWðxÞj2 ¼ ð72Þ l mx2 x2 . U0 2 This equation is the same as (18).
References [1] [2] [3] [4]
Flory PJ. Principles of polymer chemistry. Ithaca (NY): Cornell University Press; 1953. Flory PJ. Statistical mechanics of chain molecules. New York: InterScience; 1977. de Gennes PG. Scaling concepts in polymer physics. Ithaca, NY: Cornell University Press; 1979. Cannavacciuolo L, Pedersen JS. Properties of polyelectrolyte chains from analysis of angular correlation functions. J Chem Phys 2002;117:8973–82. [5] Cannavacciuolo L, Pedersen JS. Moments and distribution function of polyelectrolyte chains. J Chem Phys 2004;120:8862–5. [6] Ullner M. Comments on the scaling behavior of flexible polyelectrolytes within the Debye–Hu¨ckel approximation. J Phys Chem B 2003;107:8097–110. and references therein.