Principles of Surface Potential Estimation in Mixed Electrolyte Solutions:Taking into Account Dielectric Saturation
doi: 10.1063/1674-0068/cjcp1907132
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摘要: 基于非线性泊松-玻尔兹曼方程，推导了混合电解质溶液中考虑介电饱和度的表面电位的解析表达式.近似解析解和精确数值解计算出的表面电位在很大范围的电荷密度和离子强度条件下均具有很好的一致性.当表面电荷密度大于0.30 C/m$ ^2 $时，介电饱和度对表面电位的影响变得尤为重要；当表面电荷密度小于0.30 C/m$ ^2 $时，可忽略介电饱和度的影响，即基于经典泊松-玻尔兹曼方程可获得有效的表面电位解析模型.因此，0.3 C/m$ ^2 $可作为是否考虑介电饱和度的颗粒临界表面电荷密度值.在低表面电荷密度时，考虑介质饱和度的表面电位解析模型可自然回归到经典泊松-玻尔兹曼理论的结果，得到的表面电位可以正确地预测一价和二价反离子之间的吸附选择性.Abstract: The dielectric properties between in-particle/water interface and bulk solution are significantly different, which are ignored in the theories of surface potential estimation. The analytical expressions of surface potential considering the dielectric saturation were derived in mixed electrolytes based on the nonlinear Poisson-Boltzmann equation. The surface potentials calculated from the approximate analytical and exact numerical solutions agreed with each other for a wide range of surface charge densities and ion concentrations. The effects of dielectric saturation became important for surface charge densities larger than 0.30 C/m$ ^2 $. The analytical models of surface potential in different mixed electrolytes were valid based on original Poisson-Boltzmann equation for surface charge densities smaller than 0.30 C/m$ ^2 $. The analytical model of surface potential considering the dielectric saturation for low surface charge density can return to the result of classical Poisson-Boltzmann theory. The obtained surface potential in this study can correctly predict the adsorption selectivity between monovalent and bivalent counterions.
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Key words:
- Dielectric property /
- Electrical double layer /
- Surface charge /
- Colloid particle
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Figure 1. The relationships between ion concentration ratio and surface potential in (a) 1:1+2:2, (b) 1:1+1:2 and (c) 1:1+2:1 electrolyte solutions. The symbols represent numerical surface potential and the solid lines represent analytical surface potential, $ f_i $/$ f_j $ is the ratio of bivalent ion concentration in mixed electrolytes with 0.1 mol/L ion strength.
Figure 2. The relationships between ion concentration and surface potential in (a) 1:1+2:2, (b) 1:1+1:2 and (c) 1:1+2:1 electrolyte solutions as $ f_i $ = $ f_j $ in different surface charge densities. The symbols represent numerical surface potential and the solid lines represent analytical results.
Figure 3. Surface potential of illite particle in NaCl+CaCl$ _2 $ electrolytes, the numerical and analytical results are calculated using Eq.(26) and Eq.(27), respectively. $ K_{ \rm{ \rm{Ca/Na}}} $ (= $ f_{ \rm{Na}} $$ N_{ \rm{Ca}} $/($ f_{ \rm{Ca}} $$ N_{ \rm{Na}} $)) is selectivity coefficient, $ f_{ \rm{Na}} $ and $ f_{ \rm{Ca}} $ are concentrations of Na$ ^+ $ and Ca$ ^{2+} $ in bulk solutions, $ N_{ \rm{Na}} $ and $ N_{ \rm{Ca}} $ are adsorption amount of Na$ ^+ $ and Ca$ ^{2+} $, respectively. The surface charge density of illite particle is 0.2895 C/m$ ^2 $, the concentrations of Na$ ^+ $ and Ca$ ^{2+} $ were determined by the binary Na-Ca exchange equilibrium on illite surface [35].
Figure 4. The relationships between surface charge density and surface potential in (a) 1:1+2:2, (b) 1:1+1:2 and (c) 1:1+2:1 electrolyte solutions at different $ f_i $/$ f_j $ values with ion strength of 0.1 mol/L. The symbols represent numerical surface potential and the solid lines represent analytical results
Figure 5. The relationships between surface charge density and surface potential in the presence and absence of dielectric saturation. The ratio of bivalent ion concentration in mixed electrolytes is equal to 1, i.e. $ f_i $ = $ f_j $ = $ f $, (a) $ f $ = 0.001 mol/L and (b) $ f $ = 0.1 mol/L. The solid lines and symbols represent the analytical surface potential in the presence and absence of dielectric saturation, respectively
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