Effects of the solvation structure on diffusion of a large particle in a binary mixture studied by perturbation theory

Journal of Molecular Liquids 200 (2014) 85–88

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Journal of Molecular Liquids journal homepage: www.elsevier.com/locate/molliq

Effects of the solvation structure on diffusion of a large particle in a binary mixture studied by perturbation theory Y. Nakamura a,⁎, A. Yoshimori a, R. Akiyama b a b

Department of Physics, Kyushu University, Fukuoka 812-8581, Japan Department of Chemistry, Kyushu University, Fukuoka 812-8581, Japan

a r t i c l e

i n f o

Article history: Received 26 November 2013 Received in revised form 16 May 2014 Accepted 18 June 2014 Available online 2 July 2014 Keywords: Diffusion of biomolecule Solvation structure Perturbation expansion Stokes–Einstein relation Hard-sphere system Depletion effect

a b s t r a c t We study the effects of the solvation structure on the diffusion of a large particle in a binary mixture. Using our recently developed perturbation theory, we calculate the diffusion coefficient of a large hard-sphere solute particle immersed in a binary solvent mixture of hard spheres with two different sizes. The calculation results show that the Stokes–Einstein (SE) relation breaks down in the hard-sphere system. When the size ratio of binary solvent spheres is three or more, the deviation from the SE relation increases with the packing fraction of larger solvent spheres. In contrast, at the size ratio of two, the diffusion coefficient approaches the value predicted by the SE relation as larger solvent spheres are added. We show that the large deviation from the SE relation is caused by the high density of larger solvent spheres around the solute sphere. © 2014 Elsevier B.V. All rights reserved.

1. Introduction The diffusion of a particle larger than its surrounding liquid particles has been studied in biology as well as in physics. In particular, many researchers have carried out experiments on diffusion in a solvent mixture [1–4]. The diffusion in a mixture is related to the diffusion of proteins in a cytoplasm, which is a dense mixture of various biomolecules. Thus, it is necessary to study the diffusion in a mixture to understand biological processes in vivo, because biological phenomena such as protein transport and chemical reaction are often diffusion-limited. The diffusion of a large particle, such as a biomolecule, in a solvent mixture is affected by the solvation structure around the solute. For instance, the diffusion coefficient can be smaller when the density of solvent particles around the solute is high [5–8]. The solvation effect is more clearly seen in a solvent mixture than in a one-component solvent. Since the solvation structure changes greatly with the mixing ratio of each solvent [9–12], the diffusion coefficient can reflect a change in the solvation effect. Actually, some experiments of a mixture [1–4] have shown that the diffusion coefficient deviates from the Stokes– Einstein (SE) relation where the solvation structure is not considered. Thus, we expect that the solvation effect causes the deviation. However, the solvation effect of a mixture on diffusion is not well understood. The solvation effect of a binary mixture can be studied using a new theory recently developed by us [11,12]. Our theory does not ⁎ Corresponding author. E-mail address: [email protected] (Y. Nakamura).

http://dx.doi.org/10.1016/j.molliq.2014.06.021 0167-7322/© 2014 Elsevier B.V. All rights reserved.

suffer from the finite-size effect [13], in contrast to previous theories [14–16]. When the solute particle is large, the finite-size effect makes it difficult to calculate the diffusion coefficient using the previous theories. We avoid this difficulty by employing the perturbation expansion in powers of the size ratio between the solute and solvent particles [11,12,17]. The solvation structure can be considered through the radial distribution functions, which represent the density distribution of solvent particles around a solute particle. In the present study, by calculating the diffusion coefficient using our perturbation theory, we clarify the solvation effect of a binary mixture of solvent and cosolvent particles. The solvation effect on the diffusion coefficient is evaluated by the deviation from the value predicted by the SE relation. We investigate the dependence of the deviation on the packing fraction of cosolvent particles for various sizes of cosolvent particles. The cosolvent particles are added while maintaining a constant total packing fraction of solvent and cosolvent particles. This mixing process is in contrast to that in our previous studies [11,12]. 2. Theory 2.1. System We consider a large solute particle immersed in a binary solvent mixture of solvent 1 (solvent) and solvent 2 (cosolvent) particles. The particle radii of the solute, solvent 1, and solvent 2 are denoted by R, a, and b, respectively. We assume that the solvent 1 and 2 particles are much smaller than the solute particle (a, b ≪ R). The particle radius of

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solvent 2 is greater than or equal to the particle radius of solvent 1 (a ≤ b). In addition, all particles are electrically neutral and interact with spherically symmetric potentials. We assume that the solvent 1 and 2 particles have a small velocity u at an infinite distance from the solute fixed at the origin to obtain the diffusion coefficient through the Einstein relation. The present system is equivalent to that of a solute particle moving with velocity −u in a binary mixture. We calculate the friction exerted on the solute particle by the binary mixture, assuming the steady state. The diffusion coefficient D is given by the friction coefficient ξ through the Einstein relation D = kBT/ξ, where kB is the Boltzmann constant and T is the temperature. 2.2. Dynamics of a binary solvent mixture around a solute The dynamics of binary solvent particles around the solute can be described by the generalized Langevin equations [15,18]. When the solute particle is large, however, it is difficult to calculate the equations directly. Thus, the generalized Langevin equations are expanded in powers of the ratio between the radii of the solute and larger solvent particles, ϵ ≡ b/R. Details of the expansion were described in previous papers [11,12]. The perturbation expansion allows us to derive the hydrodynamic equations with the boundary conditions on the surface of the solute. The derived hydrodynamic equations are written as [11,12] ∇
vðrÞ ¼ 0;

ð1Þ

2

−∇δP ðrÞ þ η∇ vðrÞ ¼ 0:

ð2Þ

Here, δP(r) is the pressure, v(r) is the barycentric velocity of the binary solvent, and η is the shear viscosity. The boundary conditions on the surface of the solute particle (r = R) are derived up to the first order as [11,12] vr ðrÞ ¼ −ϵαvθ ðrÞ cot θ;

ð3Þ

where Δv j ðr Þ ¼

ω j ðr Þ ¼

2ω j ðr Þ w j ðrÞ

Z

r Rþa

 0 0 w j r dr ;

1 dw j ðr Þ ; w j ðr Þ dr

ð7Þ

ð8Þ

for j = 1 or T. Here, γ = ζ + η/3, where ζ is the bulk viscosity and wT(r) = w1(r) + w2(r). The equilibrium mass density field wi(r) is defined when u = 0. It can be calculated using the radial distribution function between the solute and solvent i, gi(r), through wi(r) = ρigi(r), where ρi is the bulk mass density of solvent i. 2.3. Friction coefficient Solving Eqs. (1) and (2) with Eqs. (3) and (4), we derive the friction coefficient ξ as ξ ¼ 8πcηR;

c¼

  3 2−ϵα þ ϵβ : 4 3 þ ðϵα Þ2 þ ϵβ

ð9Þ

ð10Þ

Here, α and β are respectively given by Eqs. (5) and (6). If the mass density field is homogeneous (wi(r) = ρi), α and β become zero. In this case, Eq. (9) is in agreement with the Stokes law, ξ = 4πηR, which gives the SE relation for the diffusion coefficient. The expression for c given by Eq. (10) includes nonlinear terms of ϵ. This expression provides more precise values than those obtained using the linearized expression for c in previous studies [11,12]. Actually, the friction coefficient calculated using the present expression (10) is in better agreement with the exact value obtained by the generalized Langevin equations. Details will be described elsewhere. 3. Model

∂vθ ðrÞ 1 þ ϵβ ϵα vθ ðrÞ− δ P ðrÞ tan θ: ¼ R 2η ∂r

ð4Þ

Here, vr(r) and vθ(r) are the r and θ components of v(r) in the spherical coordinates, respectively. The origin of the coordinate system is the center of the solute particle and the z-axis corresponds to the u direction. We assume the slip boundary condition in the zeroth order. The parameters α and β included in Eqs. (3) and (4) depend on the solvation structure through the equilibrium mass density field of solvent i, wi(r), as follows: ϵα ¼

2 Rþa

Z



∞ Rþa

 wT ðr Þ −1 dr; wT ð∞Þ

ð5Þ

and ϵβ ¼

1 Rþa

(Z

Rþb Rþa

Z Δv1 ðr Þdr þ

∞ Rþb

ΔvT ðr Þdr

 "Z Rþb Z Rþb  0   0 0 ω1 r γ w1 ðr Þ − 1þ 0 Δv1 r dr dr ð Þ w η r Rþa r 1 # Z ∞ Z ∞  0  0 0 ωT r Δv þ wT ðr Þ r dr dr T 0 Rþb r wT ðr Þ "Z  Z Rþb ∞ ω ðr Þ γ T ΔvT ðr Þdr − 1þ w1 ðr Þdr η Rþa Rþb wT ðr Þ  #) 1 1 − ; þ Δv1 ðR þ bÞ w1 ðR þ bÞ wT ðR þ bÞ

We calculate the diffusion coefficient of a large hard-sphere solute particle immersed in a binary mixture of small (solvent) and mediumsize (cosolvent) hard spheres. The radius of the solute sphere R is 50a, where a is the radius of the solvent sphere. The radius of the cosolvent sphere b varied from 2a to 5a. The mass of the cosolvent sphere is set to (b/a)3m, where m is the mass of the solvent sphere. In the present study, we mix the cosolvent and solvent spheres while maintaining a constant total packing fraction of solvent and cosolvent spheres. The total packing fraction is set at 0.38, which is almost the same as the packing fraction of water. The packing fraction of the cosolvent varied from 0.00 to 0.10. The radial distribution functions gi(r) for the present hard-sphere system are obtained using the Ornstein–Zernike integral equation coupled with the hypernetted-chain closure (the OZ-HNC theory). For the numerical calculation, we employ a hybrid convergence algorithm [19–24]. In addition, we assume γ = 0 in Eq. (6) as a first step of the study. 4. Results and discussion 4.1. Diffusion coefficient

ð6Þ

To clarify the effect of the solvation structure, we plot the diffusion coefficient ratio DSE/D in Fig. 1. Here, D is calculated using the present perturbation theory and DSE is obtained using the SE relation with the slip boundary condition [25]. Thus, the deviation of the ratio DSE/D from unity represents a breakdown of the SE relation. The breakdown of the SE relation is caused by the solvation effect since the solvation

Y. Nakamura et al. / Journal of Molecular Liquids 200 (2014) 85–88

1.5

At the cosolvent sphere radius of 5a, we obtain DSE/D = 1.29 even when the cosolvent packing fraction is as small as 0.01. Note that a decrease in DSE/D at the cosolvent sphere radius of 2a was not observed in our previous studies of a binary hard-sphere system [11,12]. For all radii of the cosolvent spheres, DSE/D increased with increasing cosolvent packing fraction from 0.00 to 0.10. Our previous studies differ from the present study in terms of the mixing process of cosolvent spheres. In the previous studies, the cosolvent spheres were added while maintaining a constant packing fraction of solvent spheres. We consider that the mixing process used in the present study is in closer agreement with experimental conditions.

DSE/D

5a 4a

1.3

87

3a 2a

4.2. Equilibrium mass density field

1.1

0

0.02

0.04

0.06

0.08

0.1

packing fraction of cosolvent Fig. 1. Dependence of the diffusion coefficient ratio DSE/D on the packing fraction of the cosolvent in a binary hard-sphere mixture. Here, D and DSE are calculated using our perturbation theory and the SE relation (slip condition), respectively. The radius of the solute sphere is 50a, where a is the radius of the solvent sphere. The radii of the cosolvent spheres are 2a (diamonds), 3a (squares), 4a (circles), and 5a (triangles).

structure is not considered in the SE relation. The SE relation is derived from the hydrodynamic equations as DSE = kBT/4πηR, where η is the shear viscosity. When the radius of the cosolvent spheres is 2a, DSE/D has a small peak at a cosolvent packing fraction of 0.01. For packing fractions smaller than 0.01, DSE/D increases with the cosolvent packing fraction. However, DSE/D decreases slowly as the cosolvent packing fraction increases from 0.01. When the cosolvent packing fraction is 0.10, DSE/D is slightly smaller than that for the pure solvent. Note that, even for the pure solvent, the diffusion coefficient is smaller than that obtained using the SE relation because of the solvation effect. When the radii of the cosolvent spheres are 3a, 4a, and 5a, the ratio DSE/D increases with increasing cosolvent packing fraction from 0.00 to 0.10. In contrast to the case of cosolvent spheres of radius 2a, a decrease in DSE/D is not observed when the cosolvent packing fraction increases from 0.00 to 0.10. Additionally, the deviation from the SE relation increases with the size of the cosolvent spheres. For instance, even when the cosolvent packing fraction is fixed at 0.10, DSE/D increases from 1.21 to 1.40 with increasing cosolvent sphere radius from 3a to 5a. When the radii of the cosolvent spheres are 4a and 5a, DSE/D increases rapidly for packing fractions smaller than 0.01. For larger packing fractions, DSE/D shows gradual growth. The increase is more rapid at the cosolvent sphere radius of 5a than that at the radius of 4a.

To reveal the origin of the breakdown of the SE relation, we plot the reduced mass density fields for the cosolvent sphere radii of 5a (Fig. 2) and a (Fig. 3). The large deviation from the SE relation is observed at the cosolvent sphere radius of 5a, while the small deviation is observed at the radius of 2a (Fig. 1). The packing fractions of the cosolvent in Figs. 2 and 3 are 0.01 and 0.09. Here, the reduced mass density is the mass density divided by the total bulk mass density ρT ≡ ρ1 + ρ2. In the present system, ρT is independent of the cosolvent packing fraction and the radius of the cosolvent spheres. When the cosolvent packing fraction is 0.09 for the cosolvent sphere radius of 5a (Fig. 2 right), we obtain a high peak of the cosolvent reduced mass density at the surface of the solute sphere (r/a = 55). The peak value is four times as large as that for the cosolvent packing fraction of 0.01. In contrast, the peak of the solvent reduced mass density does not change significantly when the cosolvent packing fraction increases from 0.01 to 0.09. Since DSE/D increases with the cosolvent packing fraction, we can relate the large deviation from the SE relation to the high peak of the cosolvent reduced mass density. When the radius of the cosolvent spheres is 2a, a low peak of the cosolvent reduced mass density is observed at the surface of the solute sphere (r/a = 52) in Fig. 3. The peak of the cosolvent reduced mass density is considerably lower than that at the cosolvent sphere radius of 5a even when the cosolvent packing fraction is 0.09. The peak value of the cosolvent reduced mass density is a major difference between the mass density fields for the cosolvent sphere radii of 2a and 5a. Thus, the low peak of the cosolvent reduced mass density is considered to be the cause of the smaller value of DSE/D. Although the peak of the cosolvent reduced mass density increases with the cosolvent packing fraction, the decrease in the peak of the solvent reduced mass density cannot be ignored at the cosolvent sphere radius of 2a. By the behavior of the reduced mass density, we explain the reason for the peak of DSE/D in the cosolvent sphere radius of 2a. We consider that the increase in DSE/D at small cosolvent packing

50 40

60 w1(r)/ρT w2(r)/ρT

w1(r)/ρ T or w2(r)/ρ T

w1(r)/ρ T or w2(r)/ρ T

60

30 20 10 0 50

52

54 r /a

56

58

50 40

w1(r)/ρ T w2(r)/ρ T

30 20 10 0 50

52

54 r /a

56

58

Fig. 2. Mass density field of the solvent (cosolvent) divided by the total bulk mass density w1(r)/ρT(w2(r)/ρT). The solid (dashed) curve represents the reduced mass density field of the solvent (cosolvent) spheres around the solute sphere. The packing fractions of the cosolvent are 0.01 (left) and 0.09 (right). The radii of the cosolvent and solute spheres are 5a and 50a, respectively, where a is the radius of the solvent sphere.

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Y. Nakamura et al. / Journal of Molecular Liquids 200 (2014) 85–88

20 w1(r)/ρT

16

w1(r)/ρ T or w2(r)/ρ T

w1(r)/ρ T or w2(r)/ρ T

20 w2(r)/ρT

12 8 4 0 50

52

54 r /a

56

58

w1(r)/ρ T w2(r)/ρ T

16 12 8 4 0 50

52

54 r /a

56

58

Fig. 3. Mass density field of the solvent (cosolvent) divided by the total bulk mass density w1(r)/ρT(w2(r)/ρT). The curves are the same as those in Fig. 2. The packing fractions of the cosolvent are 0.01 (left) and 0.09 (right). The radii of the cosolvent and solute spheres are 2a and 50a, respectively, where a is the radius of the solvent sphere.

fraction is caused by the increase in the peak of the cosolvent. In contrast, the decrease in DSE/D at larger cosolvent packing fraction than 0.01 is considered to be caused by the decrease in the peak of the solvent. From these results, we find that the breakdown of the SE relation is closely associated with the peak value of the reduced mass density fields of the solvent and cosolvent spheres. Actually, since a high peak of the reduced mass density gives large values of α and β in Eq. (10), c deviates from 1/2, which is the value used in the SE relation. The association with the peak value can be physically understood if we note that large peak values represent high mass densities of solvent and cosolvent spheres. The high mass density around the solute particle hinders its motion. Since this reduces the diffusion coefficient, we observe the breakdown of the SE relation. The high peak of the reduced mass density observed in the present study is caused by the excluded volume force [10]. 5. Concluding remarks The present calculations of the binary hard-sphere system have shown the breakdown of the SE relation due to the solvation effect. The deviation from the SE relation depends greatly on the cosolvent packing fraction and the size ratio between the solvent and cosolvent spheres. The ratio DSE/D increases with the cosolvent packing fraction when the size ratio is three or more. In contrast, DSE/D decreases at a larger cosolvent packing fraction when the size ratio is two. We found that the large deviation from the SE relation is caused by the high mass density of the cosolvent spheres around the solute. Since we maintain the constant total packing fraction of the solvent and cosolvent spheres, the addition of the cosolvent spheres changes the pressure of a mixture. As the packing fraction of the cosolvent increases from 0.00 to 0.10, the pressure decreases from Pσ 3/kBT = 4.65 to 3.27 at the cosolvent diameter of 2σ. Here, σ is the diameter of the solvent sphere and the pressure P is obtained using the Percus–Yevick approximation [26]. If we keep the pressure constant when adding the cosolvent spheres, we consider that DSE/D is a value larger than that obtained by the present study. We have found that DSE/D increases with the total packing fraction when the packing fraction of the cosolvent spheres is the same. Details will be given elsewhere. In contrast to this study of the hard sphere system, the diffusion coefficient larger than the value predicted by the SE relation has been observed in some experiments of a mixture [1,4]. We consider that this behavior is caused by other interaction systems than hard spheres.

In particular, the diffusion coefficient is considered to be larger than that by the SE relation if the cosolvent particle cannot approach the solute particle (preferential exclusion). The preferential exclusion is observed in various cosolvents, such as sucrose and polyethylene glycol, around the protein. The study of other interaction systems will be a subject for future work. Acknowledgments We thank Professor Masahiro Kinoshita of Kyoto University for the software program used in the numerical preparation to obtain the radial distribution functions. Y. N. thanks the Japan Society for the Promotion of Science for a Research Fellowship (No. 25-4159). This work was supported by Grants-in-Aid for Innovative Scientific Research Areas (20118007) and for Scientific Research C (25400428) from the Ministry of Education, Culture, Sports, Science, and Technology of Japan. References [1] M. Iwaki, T. Yanagida, Presented at Int. Symp, Hydration and ATP Energy, 2010. [2] S. Zorrilla, M.A. Hink, A.J.W.G. Visser, M.P. Lillo, Biophys. Chem. 125 (2007) 298. [3] B. Varga, F. Migliardo, E. Takacs, B. Vertessy, S. Magazu, C. Mondelli, Chem. Phys. 345 (2008) 250. [4] G.H. Koenderink, S. Sacanna, D.G.A.L. Aarts, A.P. Philipse, Phys. Rev. E. 69 (2004) 021804. [5] S. Nishida, T. Nada, M. Terazima, Biophys. J. 87 (2004) 2663. [6] S. Nishida, T. Nada, M. Terazima, Biophys. J. 89 (2005) 2004. [7] J.S. Khan, Y. Imamoto, M. Harigai, M. Kataoka, M. Terazima, Biophys. J. 90 (2006) 3686. [8] Z. Li, Phys. Rev. E. 80 (2009) 061204. [9] K. Gekko, S.N. Timasheff, Biochemistry 20 (1981) 4667. [10] Y. Karino, R. Akiyama, M. Kinoshita, J. Phys. Soc. Jpn. 78 (2009) 044801. [11] Y. Nakamura, A. Yoshimori, R. Akiyama, J. Phys. Soc. Jpn. 81 (2012) SA026. [12] Y. Nakamura, A. Yoshimori, R. Akiyama, J. Phys. Soc. Jpn. 83 (2014) 064601. [13] R.O. Sokolovskii, M. Thachuk, G.N. Patey, J. Chem. Phys. 125 (2006) 204502. [14] J.R. Mehafty, R.I. Cukier, Phys. Rev. Lett. 38 (1977) 1039. [15] T. Yamaguchi, T. Matsuoka, S. Koda, J. Chem. Phys. 123 (2005) 034504. [16] T. Yamaguchi, T. Matsuoka, S. Koda, J. Mol. Liq. 134 (2007) 1. [17] Y. Inayoshi, A. Yoshimori, R. Akiyama, J. Phys. Soc. Jpn. 81 (2012) 114603. [18] J.P. Hansen, I.R. McDonald, Theory of Simple Liquids, Academic Press, London, 1986. [19] M. Kinoshita, M. Harada, Mol. Phys. 74 (1991) 443. [20] M. Kinoshita, M. Harada, Mol. Phys. 79 (1993) 145. [21] M. Kinoshita, M. Harada, Mol. Phys. 81 (1994) 1473. [22] M. Kinoshita, D. Berard, J. Comput. Phys. 124 (1996) 230. [23] M. Kinoshita, S. Iba, K. Kuwamoto, M. Harada, J. Chem. Phys. 105 (1996) 7177. [24] M. Kinoshita, Y. Okamoto, F. Hirata, J. Comput. Chem. 19 (1998) 1724. [25] L.D. Landau, E.M. Lifshitz, Fluid Mechanics, Butterworth-Heinemann, Oxford, 1987. [26] J.L. Lebowitz, Phys. Rev. 133 (1964) A895.