平衡时溶液表面吸附

August [Article]

物理化学学报(Wuli Huaxue Xuebao) Acta Phys. -Chim. Sin. 2015, 31 (8), 1499–1503

doi: 10.3866/PKU.WHXB201506191

1499

www.whxb.pku.edu.cn

平衡时溶液的表面吸附

陈飞武* 卢天武钊

(北京科技大学化学与生物工程学院化学与化学工程系, 北京 100083;功能分子与晶态材料科学与应用北京市重点实验室, 北京 100083)

摘要: 溶液的表面吸附仍是表面热力学当中的一个具有挑战性的问题. 在本文中我们定义了一个新的热力学态函数, 表面吸附的平衡条件是这个态函数的微分为零. 基于这个条件, 我们推导了描述平衡时表面吸附的新方程. 在推导过程中没有采用假想分界面. 新的表面吸附方程和Gibbs 表面吸附方程完全不一样. 还通过分子动力学方法模拟了氯化钠溶液, 模拟结果和我们的理论预测符合较好. 关键词: 表面吸附; 平衡条件; 吉布斯吸附方程; 热力学态函数; 溶液中图分类号: O641

Surface Absorption of a Solution at Equilibrium

CHEN Fei-Wu* LU Tian WU Zhao

(Department of Chemistry and Chemical Engineering, School of Chemistry and Biological Engineering, University ofScience and Technology Beijing, Beijing 100083, P . R . China ; Beijing Key Laboratory for Science and Application of

Functional Molecular and Crystalline Materials, Beijing 100083, P . R . China ) Abstract: Surface adsorption of a solution is still a challenging problem in the thermodynamics ofsurfaces. In this work, a new thermodynamic state function is defined. The equilibrium condition of surfaceadsorption is that the differential of this state function is equal to zero. Based on this condition, we deriveda new equation to describe surface adsorption at equilibrium. No hypothetical dividing surface is needed inthis derivation. The new equation is quite different from the Gibbs adsorption equation. We also performedmolecular dynamic simulations of aqueous sodium chloride solutions. The simulated results are in goodagreement with our theoretical predictions.

Key Words: Surface absorption; Equilibrium condition; Gibbs absorption equation;

Thermodynamic state function; Solution

1 Introduction

It is well known that, for a cup of sugar water, the very thinsurface layer of the sugar water will be sweeter than its interiorpart. Though many progresses have been made so far,1–6 yet in-terpreting this interesting phenomenon quantitatively is still achallenging problem in the thermodynamics of surfaces. For amulticomponent solution, one usually start with the differentialform of the Gibbs free energy as follows7

d G =¡S d T +V d P +d ¾+

¹B d n B

(1)

where S , T , P , V , γ, σ, µB , and n B are the system's entropy, tem-perature, pressure, volume, surface tension, surface area, chem-ical potential and the number of moles of the component B, re-spectively. The third term on the right hand side of Eq.(1) is thesurface work. Since the surface area could not increase by it-self in most cases and could only be stretched out by its surr-

Received: March 4, 2015; Revised: June 19, 2015; Published on Web: June 19, 2015.*

Corresponding author. Email: [email protected].

The project was supported by the National Natural Science Foundation of China (21173020, 21473008).国家自然科学基金(21173020, 21473008)资助项目

Editorial office of Acta Physico-Chimica Sinica

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Acta Phys. -Chim. Sin. 2015

Vol.31

ounding environment, the surface work has a positive sign. Thechemical potential of the component B (µB ) in a nonelectrolytesolution has the following form¹B =¹ªB +RT ln a B

(2)

where ¹ªB and a B are the standard chemical potential and activ-ity of the component B, respectively. R is the gas constant. Foran ideal solution, the activity a B becomes equal to the concen-tration c B . For a strong electrolyte solution, which has the for-mula Mv +X v –, µB has a slightly different expression7

¹B =¹ªB +vRT ln a B

(3)

where v = v + + v –. If the surface and bulk phases are con-sidered, Eq.(1) should be expressed explicitly as

d G =¡S d T +V d P +d ¾+X ¹s B d n s B +X

¹B d n

B B

(4)

where the superscripts “s” and “α” are referred to the surface

and bulk phases, respectively.

It is well known qualitatively that the concentration c s

B in the

surface region will be bigger than the concentration c

B in the

bulk region if the surface tension decreases with c s

B , and vice versa . Since there is no term in Eq.(4) related to the change ofthe surface tension, Gibbs exploited an analog form of theGibbs-Duhem equation at constant temperature and pressure toexplain these absorption behaviors, which is

¾d +¾

X n ¾B d ¹B =0B

(5)

where n B is the surface excess amount of the component B

defined as n ¾B ´n B ¡n

B . Eq.(5) is the so-called Gibbs absorp-tion isotherm. Eq.(5) itself is still insufficient to determine therelationship between the surface tension and concentrations ofthe component B in both the surface and bulk phases because oftoo many variables involved in the equation. Therefore, someconstraint should be imposed on the surface excess amount ofsome component, usually the solvent. The dividing surface inthe surface region was such a constraint originally proposed byGibbs, which made the surface excess amount of the solventzero. Then, Gibbs derived the following absorption equation7

=¡

1@B

(6)

where Γ21, the surface excess of the component B relative to the

solvent A, is defined as

=(n s B ¡

n B

£n s A ) =¾

(7)

Eq.(6) is only valid for a two-component solution. Recently

Menger et al. 8–10 found experimentally for some systems that theright-hand side of Eq.(6) remained almost unaltered while thesurface excess Γ21 on the left hand side of Eq.(6) still changedwith the concentration of the component B, which led to the ar-guments on the Gibbs analysis.11–13

2 Theory

Based on the facts above, we started to rethink the thermody-namics of surfaces from beginning. It is well known that a

change of the concentration of a component B in the surface re-gion due to the surface absorption will lead to changes of thesurface tension and the corresponding surface work as shown inEqs.(1) and (4). Contrary to the surface works in the mostcases, this type of the surface work is not done externally by thesurrounding environment, but done by the system itself. There-fore, we think that this internal surface work should be –γd σ in-stead of γd σ as presented in Eqs.(1) and (4). This is the keystarting point in the present work. For simplicity and conveni-ence of discussions below, only Eq.(4) is rewritten as

d G =¡S d T +V d P ¡d ¾+X ¹s B d n s B +X

¹B d n

B B

(8)

If γd σ is substituted with d(γσ)–σd γ, Eq.(8) can be expressed as

d F =¡S d T +V d P +¾d +X ¹s B d n s B +X

¹B d n

B B

(9)

where F is defined asF ´G +¾

(10) As will be discussed below, the equilibrium condition of thesurface absorption is that the differential of this thermodynam-ic state function F is zero at constant temperature and pressure.From Eq.(9), the differential form of the chemical potential inthe surface phase can be derivedd ¹s B =¡S B ; m d T +V B ; m d P +¾B ; m d

(11)

where S B, m , V B, m , and σB, m are the partial molar entropy, partialmolar volume, and partial molar surface area, respectively. Incomparison with Eq.(8), the third term on the right hand side ofEq.(9) is directly related to the change of the surface tension, aswe expect. The total differential form of the surface tension iswritten as

d =X B

µ@B

¶d n s C =B

B (12) provided that the temperature and pressure remain constant.Substituting Eq.(12) into Eq.(9) leads to

d F =¡S d T +V d P +X µ¾µ@@n ¶

B C =B ¹B ¶B

(13) s d n s B +X ¹B d n B

Since the total amount of moles of the component B, n B =

n s B +n

B , in the surface and bulk regions are fixed, therefore

d n B =¡d n s

B . With this equality Eq.(13) becomes

d F =¡S d T +V d P +X µ¾µ@¶

+¹s B ¡¹

B ¶d n s C =B

B B

(14) At constant temperature and pressure the equilibrium conditiond F = 0 results in the following equation

¾µ@¶+¹s B C =B B =¹B (15) It is seen clearly from Eq.(15) that the chemical potentials in

No.8

CHEN Fei-Wu et al.: Surface Absorption of a Solution at Equilibrium

1501

the surface and bulk phases are not equal. Substituting the ex-s

pressions of ¹B and ¹B in the Eq.(2) or Eq.(3) into Eq.(15), wefinally obtain

µ¶@a B ¾=³RT ln (16) @n B C =B a B where ζ will be 1 or v if the solute is a nonelectrolyte or electro-lyte, respectively. In the above derivation, the standard chemic-al potentials in the surface and bulk phases are considered to be

equal. It is shown from Eq.(16) that a B will be bigger than a B if

the derivative of the surface tension with n B is negative, andvice versa. This is in accordance with the surface absorption be-havior of the component B in a solution.

If the solution is very dilute the chemical potentials of thesolvent in the surface and bulk phases can be regarded to be ap-proximately equal. Then we get another equation to describethe relationship between the surface tension and the chemicalpotentials of the solute, i.e.,

¾B ; m (¡0) +¹s B =¹B

two 4 nm thick vacuum layers. Therefore the final size of simu-lation box is 4 nm × 4 nm × 16 nm. Gromacs program17,18 was

employed for simulations at constant volume and temperature.The temperature was maintained at 300 K via Nosé-Hooverthermostat. 19,20 Kirkwood-Buff force field21,22 and SPC/E model23were used to represent NaCl and water, respectively. The watergeometry was constrained with SETTLE technique.24 Longrange electrostatic interactions were evaluated by the ParticleMesh Ewald (PME) approach,25 and van der Waals interactionswere truncated at the cut-off distance of 0.14 nm. The surfacetensions were calculated by26

¿À11

=L z P zz ¡(P x x +P yy ) (19) 22where L z is the length of the box in the z direction which is nor-mal to the surface, the

P xx , P yy , and P zz are the diagonal com-ponents of the pressure tensor.

Ten ionic concentration distribution curves are presented in

Fig.1. The thickness of a surface layer is determined as the dis-tance at z direction with the density of NaCl starting from zeroto the density in bulk. As can be seen from the figure, the sur-face layers of the above systems are all approximately 0.8 nmthick. One snapshot of molecular dynamic trajectory is presen-ted in Fig.2 to illustrate the distributions of NaCl in solutionduring the simulation. It is a side view of the whole simulationbox.

(17)

provided that σB, m remains approximately constant. γ0 is the sur-face tension of the pure solvent. Substituting the expressions of

¹s B and ¹B in Eq.(2) or Eq.(3) into Eq.(17) leads to

³RT a s

=0¡ln B

¾B ; m a B

(18)

Eqs.(17) and (18) are valid only for a two-component solution.

Eq.(18) has also been derived and discussed previously byNath, 14 Li15 and Yu16 et al.

3 Results and discussion

In order to test the validity of Eq.(16), molecular dynamicsimulations of aqueous sodium chloride solutions have beenperformed. As will be clear below, the reason to chooseaqueous sodium chloride solutions is that these solutions havesimilar absorption behaviors as observed by Menger et al. 8–10First a rectangular box with dimensions of 4 nm × 4 nm × 8 nmwas set up and about 4200 water molecules were filled into thecenter of the box to yield a 8 nm-thick water layer. Then somewater molecules were replaced with Na+ and Cl– ions. Totally10 systems with NaCl concentrations ranging from 0.2 to 2.0mol L –1 with increment step of around 2.0 molL –1 were invest-igated. The simulation box was extended in both sides to yield

Fig.1 Ten density profiles (ranging from 0.2 to 2.0 mol∙L–1 with incre-mental size of 2.0 mol∙L–1) of NaCl with respect to the distance in

z direction

N : number density of Nacl pairs

¢¢

Fig.2 Snapshot of molecular dynamic trajectory in simulation

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Acta Phys. -Chim. Sin. 2015

Vol.31

Fig.3 Simulated surface tension versus the number of moles of NaCl in

the surface region

The plot of the simulated surface tension (γ) versus the num-ber of moles of NaCl in the surface region (n s

NaCl ) is shown inFig.3. It can be seen from Fig.3 that the surface tension in-creases as n s

NaCl becomes larger. These data were then fitted to

a straight line: =0:405¢

n s

NaCl +58:607. The interceptionvalue of 58.607 × 10–3 Nm –1 corresponds to the pure water sur-face tension. Though it is in good agreement with recent mo-lecular dynamic simulation,26 yet the simulated surface tensionof pure SPC/E water is lower than the experimental value of71.6 × 10–3 N¢

m –1 because no long-range dispersion correctionis included. Linear correlation coefficient and root mean squaredeviation of the fitting are 0.984 and 0.277 × 10–3 N¢

m –1, re-spectively. This linear behavior in the sodium chloride solutionwas also observed in the other research works.26–28 Because ofthe good linear correlation of the simulated data and also thedifficulty to calculate the derivative @=@n s NaCl by the molecu-lar dynamic simulation, the derivative value of 0.405 × 1021N ¢m –1¢

mol –1 is directly taken from the fitting and will be usedbelow to calculate the gas constant R in Eq.(16).

The plot of the simulated concentration of NaCl in the bulkregion versus the concentration of NaCl in the surface region is

shown in Fig.4. These data were also fitted to a line c

NaCl =2:4806c s NaCl the root mean square deviation are 0.99602 and 0.06454, re-¡0:0590. The linear correlation coefficient and

spectively. The ratio of

c NaCl /c s

NaCl can be regarded approxim-ately to be 2.4806 because of the interception value (–0.0590)

Fig.4 Simulated concentration of NaCl in the bulk region versus

concentration of NaCl in the surface region

of the fitted line with the axis of c

NaCl close to zero. This ratio

is also considered approximately as the value of a NaCl =a s

NaCl . The reason that the fitted line in Fig.4 passes slightly awayfrom the original point is perhaps due to the numerical noise ofmolecular dynamic simulation.

If we substitute @=@n s NaCl = 0.405 × 1021 N¢m

–1

mol –1 andthe ratio a NaCl =a s

NaCl = 2.4806 together with surface area σ =1.6 × 10–17 m2 and the temperature T = 300 K into Eq.(16), weobtain the gas constant R to be 11.9 JK –1mol –1. Though thecomputed ¢ R is¢

slightly larger than the¢ experimental¢

value(8.314 JK –1mol –1), yet they are at the same order of mag-nitude. It is possible that a better accuracy of the computed R would be achieved if improved force fields for sodium chlorideand water were considered. Nevertheless, the preliminary res-ults presented here are promising and satisfactory.

On the other hand, the derivative of the surface tension γ with

respect to ln c s

NaCl in the surface phase on right hand side ofEq.(6) is close to a constant, but the surface excess Γ21 on the

left hand side of Eq.(6) still changes with c s

NaCl . This is similarto the experimental observations made by Menger et al. 8–10Therefore, it is expected that Eq.(16) may be exploited to solvethe problems in the works of Menger

et al. 4 Conclusions

A new thermodynamic state function F is defined to de-scribe the thermodynamics of surfaces. The equilibrium condi-tion of the surface absorption of a solution is that dF = 0. Basedon this, a new absorption equations such as Eq.(16) are derived.Molecular dynamic similations of aqueous solutions of sodiumchloride are in good agreement with our theoretical analysis. In-stead of Gibbs absorption equation, it is hopeful that Eq.(16)may be a promising alternative to solve the problems found byMenger et al. 8–10

Acknowledgment: The authors are very grateful to Prof.LI Le-Min of Peking Univeristy for his valuable discussion.References

(1)Cheng, X. H.; Zhao, O. D.; Zhao, H. N.; Huang, J. B. Acta

Phys. -Chim. Sin. 2014, 30, 917. [程新皓, 赵欧狄, 赵海娜, 黄建滨. 物理化学学报, 2014, 30, 917.] doi: 10.3866/PKU.WHXB201403191

(2)Hu, S. Q.; Ji, X. J; Fan, Z. Y.; Zhang, T. T.; Sun, S. Q. Acta

Phys. -Chim. Sin. 2015, 31, 83. [胡松青, 纪贤晶, 范忠钰, 张田田, 孙霜青, 物理化学学报, 2015, 31, 83.] doi: 10.3866/PKU.WHXB201411191

(3)Wang, K.; Yu, Y. X.; Gao, G. H. J. Chem. Phys. 2008, 128,

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(5)Ghosh, S.; Roy, A.; Banik, D.; Kundu, N.; Kuchlyan, J.; Dhir, A.;

Sarkar, N. Langmuir 2015, 31, 2310. doi: 10.1021/la504819v

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(6)Bera, M. K.; Antonio, M. R. Langmuir 2015, 31, 5432.

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J. R. Langmuir 2013, 29, 9324. doi: 10.1021/la4018344(14)Nath, S. J. Colloid Interface Sci. 1999, 209, 116.

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August [Article]

物理化学学报(Wuli Huaxue Xuebao) Acta Phys. -Chim. Sin. 2015, 31 (8), 1499–1503

doi: 10.3866/PKU.WHXB201506191

1499

www.whxb.pku.edu.cn

平衡时溶液的表面吸附

陈飞武* 卢天武钊

(北京科技大学化学与生物工程学院化学与化学工程系, 北京 100083;功能分子与晶态材料科学与应用北京市重点实验室, 北京 100083)

Surface Absorption of a Solution at Equilibrium

CHEN Fei-Wu* LU Tian WU Zhao

Key Words: Surface absorption; Equilibrium condition; Gibbs absorption equation;

Thermodynamic state function; Solution

1 Introduction

d G =¡S d T +V d P +d ¾+

¹B d n B

(1)

Received: March 4, 2015; Revised: June 19, 2015; Published on Web: June 19, 2015.*

Corresponding author. Email: [email protected].

The project was supported by the National Natural Science Foundation of China (21173020, 21473008).国家自然科学基金(21173020, 21473008)资助项目

Editorial office of Acta Physico-Chimica Sinica

1500

Acta Phys. -Chim. Sin. 2015

Vol.31

ounding environment, the surface work has a positive sign. Thechemical potential of the component B (µB ) in a nonelectrolytesolution has the following form¹B =¹ªB +RT ln a B

(2)

¹B =¹ªB +vRT ln a B

(3)

where v = v + + v –. If the surface and bulk phases are con-sidered, Eq.(1) should be expressed explicitly as

d G =¡S d T +V d P +d ¾+X ¹s B d n s B +X

¹B d n

B B

(4)

where the superscripts “s” and “α” are referred to the surface

and bulk phases, respectively.

It is well known qualitatively that the concentration c s

B in the

surface region will be bigger than the concentration c

B in the

bulk region if the surface tension decreases with c s

¾d +¾

X n ¾B d ¹B =0B

(5)

where n B is the surface excess amount of the component B

defined as n ¾B ´n B ¡n

=¡

1@B

(6)

where Γ21, the surface excess of the component B relative to the

solvent A, is defined as

=(n s B ¡

n B

£n s A ) =¾

(7)

Eq.(6) is only valid for a two-component solution. Recently

2 Theory

Based on the facts above, we started to rethink the thermody-namics of surfaces from beginning. It is well known that a

d G =¡S d T +V d P ¡d ¾+X ¹s B d n s B +X

¹B d n

B B

(8)

If γd σ is substituted with d(γσ)–σd γ, Eq.(8) can be expressed as

d F =¡S d T +V d P +¾d +X ¹s B d n s B +X

¹B d n

B B

(9)

where F is defined asF ´G +¾

(11)

d =X B

µ@B

¶d n s C =B

B (12) provided that the temperature and pressure remain constant.Substituting Eq.(12) into Eq.(9) leads to

d F =¡S d T +V d P +X µ¾µ@@n ¶

B C =B ¹B ¶B

(13) s d n s B +X ¹B d n B

Since the total amount of moles of the component B, n B =

n s B +n

B , in the surface and bulk regions are fixed, therefore

d n B =¡d n s

B . With this equality Eq.(13) becomes

d F =¡S d T +V d P +X µ¾µ@¶

+¹s B ¡¹

B ¶d n s C =B

B B

(14) At constant temperature and pressure the equilibrium conditiond F = 0 results in the following equation

¾µ@¶+¹s B C =B B =¹B (15) It is seen clearly from Eq.(15) that the chemical potentials in

No.8

CHEN Fei-Wu et al.: Surface Absorption of a Solution at Equilibrium

1501

the surface and bulk phases are not equal. Substituting the ex-s

pressions of ¹B and ¹B in the Eq.(2) or Eq.(3) into Eq.(15), wefinally obtain

equal. It is shown from Eq.(16) that a B will be bigger than a B if

the derivative of the surface tension with n B is negative, andvice versa. This is in accordance with the surface absorption be-havior of the component B in a solution.

¾B ; m (¡0) +¹s B =¹B

two 4 nm thick vacuum layers. Therefore the final size of simu-lation box is 4 nm × 4 nm × 16 nm. Gromacs program17,18 was

¿À11

=L z P zz ¡(P x x +P yy ) (19) 22where L z is the length of the box in the z direction which is nor-mal to the surface, the

P xx , P yy , and P zz are the diagonal com-ponents of the pressure tensor.

Ten ionic concentration distribution curves are presented in

(17)

provided that σB, m remains approximately constant. γ0 is the sur-face tension of the pure solvent. Substituting the expressions of

¹s B and ¹B in Eq.(2) or Eq.(3) into Eq.(17) leads to

³RT a s

=0¡ln B

¾B ; m a B

(18)

Eqs.(17) and (18) are valid only for a two-component solution.

Eq.(18) has also been derived and discussed previously byNath, 14 Li15 and Yu16 et al.

3 Results and discussion

Fig.1 Ten density profiles (ranging from 0.2 to 2.0 mol∙L–1 with incre-mental size of 2.0 mol∙L–1) of NaCl with respect to the distance in

z direction

N : number density of Nacl pairs

¢¢

Fig.2 Snapshot of molecular dynamic trajectory in simulation

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Acta Phys. -Chim. Sin. 2015

Vol.31

Fig.3 Simulated surface tension versus the number of moles of NaCl in

the surface region

The plot of the simulated surface tension (γ) versus the num-ber of moles of NaCl in the surface region (n s

NaCl ) is shown inFig.3. It can be seen from Fig.3 that the surface tension in-creases as n s

NaCl becomes larger. These data were then fitted to

a straight line: =0:405¢

n s

m –1 because no long-range dispersion correctionis included. Linear correlation coefficient and root mean squaredeviation of the fitting are 0.984 and 0.277 × 10–3 N¢

mol –1 is directly taken from the fitting and will be usedbelow to calculate the gas constant R in Eq.(16).

The plot of the simulated concentration of NaCl in the bulkregion versus the concentration of NaCl in the surface region is

shown in Fig.4. These data were also fitted to a line c

NaCl =2:4806c s NaCl the root mean square deviation are 0.99602 and 0.06454, re-¡0:0590. The linear correlation coefficient and

spectively. The ratio of

c NaCl /c s

NaCl can be regarded approxim-ately to be 2.4806 because of the interception value (–0.0590)

Fig.4 Simulated concentration of NaCl in the bulk region versus

concentration of NaCl in the surface region

of the fitted line with the axis of c

NaCl close to zero. This ratio

is also considered approximately as the value of a NaCl =a s

NaCl . The reason that the fitted line in Fig.4 passes slightly awayfrom the original point is perhaps due to the numerical noise ofmolecular dynamic simulation.

If we substitute @=@n s NaCl = 0.405 × 1021 N¢m

–1

mol –1 andthe ratio a NaCl =a s

NaCl = 2.4806 together with surface area σ =1.6 × 10–17 m2 and the temperature T = 300 K into Eq.(16), weobtain the gas constant R to be 11.9 JK –1mol –1. Though thecomputed ¢ R is¢

slightly larger than the¢ experimental¢

On the other hand, the derivative of the surface tension γ with

respect to ln c s

NaCl in the surface phase on right hand side ofEq.(6) is close to a constant, but the surface excess Γ21 on the

left hand side of Eq.(6) still changes with c s

NaCl . This is similarto the experimental observations made by Menger et al. 8–10Therefore, it is expected that Eq.(16) may be exploited to solvethe problems in the works of Menger

et al. 4 Conclusions

Acknowledgment: The authors are very grateful to Prof.LI Le-Min of Peking Univeristy for his valuable discussion.References

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