3365:
120:
4703:
5102:
318:
4385:. These expressions are useful for analyzing quantum transport in a mesoscopic system. In metal-insulator semiconductor tunneling junctions, the electrons can build up close to the interface between layers and as a result the quantum transport of the system will be affected by the electron-electron interactions. Certain transport properties such as
4397:
5146:
reduced surface potential. On the other hand, including the finite size of the ions causes the opposite effect. The
Poisson–Boltzmann equation is most appropriate for approximating the electrostatic potential at the surface for aqueous solutions of univalent salts at concentrations smaller than 0.2 M and potentials not exceeding 50–80 mV.
4852:
3361:, where the decay is steeper than exponential decay. The following figure employs the linearized equation and the high potential graphing equation derived above. It is a potential-versus-distance graph for varying surface potentials of 50, 100, 150, and 200 mV. The equations employed in this figure assume an 80mM NaCl solution.
5138:
charges, where ions were assumed to interact with the average electrostatic field of all their neighbors rather than each neighbor individually. In addition, non-Coulombic interactions were not considered and certain interactions were unaccounted for, such as the overlap of ion hydration spheres in an aqueous system. The
3282:
137:
3747:
3390:
The
Poisson–Boltzmann equation can be applied to biomolecular systems. One example is the binding of electrolytes to biomolecules in a solution. This process is dependent upon the electrostatic field generated by the molecule, the electrostatic potential on the surface of the molecule, as well as the
1200:
The geometry that most easily facilitates this is a planar surface. In the case of an infinitely extended planar surface, there are two dimensions in which the potential cannot change because of symmetry. Assuming these dimensions are the y and z dimensions, only the x dimension is left. Below is the
4707:
Applying the equation above to the MIS tunneling junction, electronic transport can be analyzed along the z-axis, which is referenced perpendicular to the plane of the layers. An n-type junction is chosen in this case with a bias V applied along the z-axis. The self-consistent average potential of
1824:
and that is generally the standard. Some boundary conditions that apply in low potential cases are that: at the surface, the potential must be equal to the surface potential and at large distances from the surface the potential approaches a zero value. This distance decay length is yielded by the
5137:
As with any approximate model, the
Poisson–Boltzmann equation is an approximation rather than an exact representation. Several assumptions were made to approximate the potential of the diffuse layer. The finite size of the ions was considered negligible and ions were treated as individual point
3377:
The
Poisson–Boltzmann equation can be applied in a variety of fields mainly as a modeling tool to make approximations for applications such as charged biomolecular interactions, dynamics of electrons in semiconductors or plasma, etc. Most applications of this equation are used as models to gain
5145:
Though the model faces certain limitations, it describes electric double layers very well. The errors resulting from the previously mentioned assumptions cancel each other for the most part. Accounting for non-Coulombic interactions increases the ion concentration at the surface and leads to a
4393:
can be known by solving for self-consistent
Coulombic average potential from the electron-electron interactions, which is related to electronic distribution. Therefore, it is essential to analytically solve the Poisson–Boltzmann equation in order to obtain the analytical quantities in the MIS
655:
The equation for local ion density can be substituted into the
Poisson equation under the assumptions that the work being done is only electric work, that our solution is composed of a 1:1 salt (e.g., NaCl), and that the concentration of salt is much higher than the concentration of ions. The
3356:
In low potential cases, the high potential equation may be used and will still yield accurate results. As the potential rises, the low potential, linear case overestimates the potential as a function of distance from the surface. This overestimation is visible at distances less than half the
1683:. The high-potential case becomes more complex so if applicable, use the low-potential equation. In the low-potential condition, the linearized version of the Poisson–Boltzmann equation (shown below) is valid, and it is commonly used as it is more simple and spans a wide variety of cases.
2731:
2959:
23:
describes the distribution of the electric potential in solution in the direction normal to a charged surface. This distribution is important to determine how the electrostatic interactions will affect the molecules in solution. The
Poisson–Boltzmann equation is derived via
4394:
tunneling junctions. Applying the following analytical solution of the
Poisson–Boltzmann equation (see section 2) to MIS tunneling junctions, the following expression can be formed to express electronic transport quantities such as electronic density and electric current
3846:
2214:
4698:{\displaystyle f_{1}f^{0}-f_{0}+{\frac {eE_{z}\tau _{0}}{m}}{\frac {\partial f_{0}}{\partial v_{z}}}\left(1-e^{\frac {-\tau }{\tau _{0}}}\right)-\int _{0}^{t}{\frac {e}{m}}e{^{\frac {t-\tau '}{\tau _{0}}}}\nabla \rho \times {\frac {\partial f_{0}}{\partial v}}dt'}
3402:
in an ionic solution with different number of bound ions at varying physiological ionic strengths. It is shown that electrostatic potential depends on the charge of the molecule, while the electrostatic free energy takes into account the net charge of the system.
5097:{\displaystyle \rho _{2}\approx {\frac {ne{\sqrt {\pi }}G(i\lambda _{D1})e^{{\frac {-t}{\tau _{0}}}-\lambda _{D1}z}}{3{\sqrt {3}}\varepsilon _{0}\varepsilon _{r}\lambda _{D1}}}\left(1-e^{1-{\sqrt {\frac {2ne^{2}t^{2}}{m\varepsilon _{0}\varepsilon _{r}}}}}\right)}
3038:
1512:
1941:
4345:
3539:
1394:
2223:
The high-potential case is referred to as the “full one-dimensional case”. In order to obtain the equation, the general solution to the
Poisson–Boltzmann equation is used and the case of low potentials is dropped. The equation is solved with a
2462:
5117:
The electronic density and electric current can be found by manipulation to equation 16 above as functions of position z. These electronic transport quantities can be used to help understand various transport properties in the system.
4119:
116:-like qualities of the electric double layer. A simple planar case with a negatively charged surface can be seen in the figure below. As expected, the concentration of counter-ions is higher near the surface than in the bulk solution.
2506:
4845:
2740:
3531:
The electrostatic free energy can also be expressed by taking the process of the charging system. The following expression utilizes chemical potential of solute molecules and implements the
Poisson-Boltzmann Equation with the
4017:
4264:
313:{\displaystyle \nabla ^{2}\psi ={\frac {\partial ^{2}\psi }{\partial x^{2}}}+{\frac {\partial ^{2}\psi }{\partial y^{2}}}+{\frac {\partial ^{2}\psi }{\partial z^{2}}}=-{\frac {\rho _{e}}{\varepsilon _{r}\varepsilon _{0}}},}
3503:
1159:
The Poisson–Boltzmann equation can take many forms throughout various scientific fields. In biophysics and certain surface chemistry applications, it is known simply as the Poisson–Boltzmann equation. It is also known in
524:
2078:
3921:
928:
3757:
1788:
829:
2114:
1635:
1147:
4377:
such as a mesoscopic system. This is done by solving the Poisson–Boltzmann equation analytically in the three-dimensional case. Solving this results in expressions of the distribution function for the
5142:
of the solvent was assumed to be constant, resulting in a rough approximation as polar molecules are prevented from freely moving when they encounter the strong electric field at the solid surface.
5481:
1176:. Only minor modifications are necessary to apply the Poisson–Boltzmann equation to various interfacial models, making it a highly useful tool in determining electrostatic potential at surfaces.
3347:
1401:
1732:
3011:
2276:
1860:
4272:
1987:
1206:
2735:
In order to obtain a more useful equation that facilitates graphing high potential distributions, take the natural logarithm of both sides and solve for the dimensionless potential, y.
2287:
1557:
1822:
4749:
1398:
Analytical solutions have also been found for axial and spherical cases in a particular study. The equation is in the form of a logarithm of a power series and it is as follows:
4357:, and membranes. This involves the equations being solved with simple boundary conditions such as constant surface potential. These approximations are useful in fields such as
619:
410:
377:
1991:
As salt concentration increases, the Debye length decreases due to the ions in solution screening the surface charge. A special instance of this equation is for the case of
2109:
is the salt concentration in mol/L. These equations all require 1:1 salt concentration cases, but if ions that have higher valence are present, the following case is used.
733:
2019:
1853:
694:
3277:{\displaystyle \psi ={\frac {2k_{B}T}{e}}\cdot \ln {\frac {e^{y_{0}/2}+1+(e^{y_{0}/2}-1)\cdot e^{-\mathrm {K} x}}{e^{y_{0}/2}+1-(e^{y_{0}/2}-1)\cdot e^{-\mathrm {K} x}}}}
5149:
In the limit of strong electrostatic interactions, a strong coupling theory is more applicable than the weak coupling assumed in deriving the Poisson-Boltzmann theory.
2282:
and the boundary conditions that at large distances from the surface, the dimensionless potential and its derivative are zero, the high potential equation is revealed.
559:
348:
3430:
The Poisson–Boltzmann equation can also be used to calculate the electrostatic free energy for hypothetically charging a sphere using the following charging integral:
2499:
3526:
1665:
3031:
2107:
588:
4024:
437:
3742:{\displaystyle \Delta G^{\text{el}}=\int _{V}\left(kT\sum _{i}c_{i}^{\infty }\left+p^{f}U-{\frac {-\varepsilon ({\boldsymbol {\nabla }}U)^{2}}{8\pi }}\right)dV}
645:
2026:
3422:
layers of the erythrocyte membrane. This information is useful for many reasons including the study of the mechanical stability of the erythrocyte membrane.
81:
Surface charge neutralized by a molecular layer of counter-ions; surface charge potential linearly dissipated from surface to counter-ions to satisfy charge
57:. Due to thermal motion of ions, the layer of counter-ions is a diffuse layer and is more extended than a single molecular layer, as previously proposed by
3433:
4757:
3368:
Potential versus distance for varying surface potentials of 50, 100, 150, and 200 mV. The equations employed in this figure assume an 80mM NaCl solution.
3928:
738:
4126:
944:
3406:
Another example of utilizing the Poisson–Boltzmann equation is the determination of an electric potential profile at points perpendicular to the
5762:
Zhang Li-Zhi; Wang Zheng-Chuan (2009). "Analytical Solution to the Boltzmann-Poisson Equation and Its Application to MIS Tunneling Junctions".
5268:
Fogolari, F.; Brigo, A.; Molinari, H. (2002). "The Poisson–Boltzmann Equation for Biomolecular Electrostatics: a Tool for Structural Biology".
4269:
Finally, by combining the last three term the following equation representing the outer space contribution to the free energy density integral
2726:{\displaystyle e^{y/2}={\frac {e^{y_{0}/2}+1+(e^{y_{0}/2}-1)\cdot e^{-\mathrm {K} x}}{e^{y_{0}/2}+1-(e^{y_{0}/2}-1)\cdot e^{-\mathrm {K} x}}}}
2954:{\displaystyle y=2\ln {\frac {e^{y_{0}/2}+1+(e^{y_{0}/2}-1)\cdot e^{-\mathrm {K} x}}{e^{y_{0}/2}+1-(e^{y_{0}/2}-1)\cdot e^{-\mathrm {K} x}}}}
1790:; however, the results that the equations yields are valid for a wider range of potentials, from 50–80mV. Nevertheless, at room temperature,
103:
Finite ion size and hydration sphere considered; some ions are specifically adsorbed by the surface in the plane, known as the Stern layer
3855:
834:
3288:
1744:
4369:
An analytical solution to the Poisson–Boltzmann equation can be used to describe an electron-electron interaction in a metal-insulator
5239:
1947:
1562:
941:
Substituting these Boltzmann relations into the local electric charge density expression, the following expression can be obtained
5909:
3754:
The above expression can be rewritten into separate free energy terms based on different contributions to the total free energy
28:
assumptions. From the Poisson–Boltzmann equation many other equations have been derived with a number of different assumptions.
4711:
5586:
5351:
5249:
1671:
of negative ions in the zero potential region). For the spherical case, L=2, the axial case, L=1, and the planar case, L=0.
50:
5499:
D’Yachkov, L. G. (2005). "Analytical Solution of the Poisson–Boltzmann Equation in Cases of Spherical and Axial Symmetry".
3841:{\displaystyle \Delta G^{\text{el}}=\Delta G^{\text{ef}}+\Delta G^{\text{em}}+\Delta G^{\text{mob}}+\Delta G^{\text{solv}}}
2209:{\displaystyle \mathrm {K} ={\sqrt {{\frac {e^{2}}{\varepsilon \varepsilon _{0}k_{\mathrm {B} }T}}\sum c_{i}{Z_{i}}^{2}}}}
5672:
3364:
1686:
2966:
2231:
5542:
Tuinier, R. (2003). "Approximate Solutions to the Poisson–Boltzmann Equation in Spherical and Cylindrical Geometry".
53:, a charged solid comes into contact with an ionic solution, creating a layer of surface charges and counter-ions or
2279:
1151:
Finally the charge density can be substituted into the Poisson equation to produce the Poisson–Boltzmann equation.
5177:
Netz, R.R.; Orland, H. (2000-02-01). "Beyond Poisson-Boltzmann: Fluctuation effects and correlation functions".
1519:
84:
Thermal motion, ion diffusion, adsorption onto the surface, solvent/surface interactions considered negligible
1793:
61:
in the Helmholtz model. The Stern Layer model goes a step further and takes into account the finite ion size.
1185:
735:
respectively. These work equations can be substituted into the Boltzmann equation, producing two expressions
5425:"The Application of a Dynamic Stern Layer Model to Electrophoretic Mobility Measurements of Latex Particles"
1679:
When using the Poisson–Boltzmann equation, it is important to determine if the specific case is low or high
5919:
1507:{\displaystyle {\frac {d^{2}\psi }{dr^{2}}}+{\frac {L}{r}}{\frac {d\psi }{dr}}=e^{\psi }-\delta e^{-\psi }}
5914:
5896:
and Robert M. Strain, 2009, University of Pennsylvania, Department of Mathematics, Philadelphia, PA, USA.
1936:{\displaystyle \mathrm {K} ={\sqrt {\frac {2c_{0}e^{2}}{\varepsilon \varepsilon _{0}k_{\mathrm {B} }T}}}}
1201:
Poisson–Boltzmann equation solved analytically in terms of a second order derivative with respect to x.
5158:
4340:{\displaystyle \Delta G^{\text{out}}=\Delta G^{\text{em}}+\Delta G^{\text{mob}}+\Delta G^{\text{solv}}}
54:
1389:{\displaystyle {\frac {d^{2}\psi }{dx^{2}}}={\frac {c_{0}e}{\varepsilon \varepsilon _{0}}}\cdot \left}
595:
388:
355:
5477:"Recent Progress in Numerical Methods for the Poisson-Boltzmann Equation in Biophysical Applications"
5448:
128:
5371:
2457:{\displaystyle e^{-\mathrm {K} x}={\frac {(e^{y/2}-1)(e^{y_{0}/2}+1)}{(e^{y/2}+1)(e^{y_{0}/2}-1)}}}
1559:
and the lengths are measured in units of the Debye electron radius in the region of zero potential
699:
131:
of ions in the diffuse layer. The three-dimensional potential distribution can be described by the
5863:– A free, open-source Poisson-Boltzmann electrostatics and biomolecular solvation software package
1994:
1831:
663:
3395:
1165:
2225:
532:
326:
5476:
2469:
380:
46:
5805:
Moreira, A. G.; Netz, R. R. (2000). "Strong-coupling theory for counter-ion distributions".
5424:
4114:{\displaystyle \Delta G^{\text{mob}}=kT\int _{V}c_{i}\ln {\frac {c_{i}}{c_{i}^{\infty }}}dV}
3508:
1640:
5824:
5771:
5729:
5622:
5551:
5508:
5196:
3016:
2085:
566:
427:
5422:
590:
is the work required to move an ion closer to the surface from an infinitely far distance,
8:
5613:
5828:
5775:
5733:
5626:
5555:
5512:
5200:
95:
Finite ion size ignored; uniformly-charged surface; non-Coulombic interactions ignored
5840:
5814:
5787:
5693:
5667:
5643:
5608:
5524:
5396:
5293:
5220:
5186:
4390:
4378:
4374:
2278:, which is not to be confused with the spatial coordinate symbol, y. Employing several
1680:
1189:
630:
622:
431:
419:
42:
5634:
5607:
Fogolari, Federico; Zuccato, Pierfrancesco; Esposito, Gennaro; Viglino, Paola (1999).
5563:
5866:
5791:
5783:
5698:
5648:
5582:
5528:
5347:
5285:
5245:
5224:
5212:
4358:
1169:
656:
electric work to bring a charged cation or charged anion to a surface with potential
58:
25:
5844:
5742:
5717:
5297:
4840:{\displaystyle \rho _{1}\approx {\frac {aE_{z}}{2\lambda _{D1}}}e^{-\lambda _{D1}z}}
5889:
5832:
5779:
5737:
5688:
5680:
5638:
5630:
5559:
5516:
5277:
5204:
4386:
4382:
1161:
132:
4012:{\displaystyle \Delta G^{\text{em}}=\int _{V}{\frac {\sum _{i}c_{i}z_{i}qU}{2}}dV}
106:
Stern layer is thin compared to particle size; fluid velocity = 0 in Stern layer
5893:
5890:
Global classical solutions of the Boltzmann equation with long-range interactions
4349:
These equations can act as simple geometry models for biological systems such as
4259:{\displaystyle \Delta G^{\text{solv}}=kT\int _{V}\sum _{i}c_{i}^{\infty }\leftdV}
5836:
3533:
3379:
1668:
16:
Equation used for physiological interfaces, polymer science, and semiconductors
5903:
5878:
5718:"Solutions of non-linear Poisson–Boltzmann equation for erythrocyte membrane"
5241:
Thermodynamics and Statistical Mechanics: Equilibrium by Entropy Maximisation
5216:
4370:
4354:
3407:
5716:
Cruz, Frederico A. O.; Vilhena, Fernando S. D. S.; Cortez, Celia M. (2000).
5609:"Biomolecular Electrostatics with the Linearized Poisson–Boltzmann Equation"
119:
5702:
5289:
5139:
5111:
3358:
1826:
519:{\displaystyle c_{i}=c_{i}^{0}\cdot e^{\frac {-W_{i}}{k_{\mathrm {B} }T}},}
41:
The Poisson–Boltzmann equation describes a model proposed independently by
5652:
5208:
5819:
5191:
4373:(MIS). This can be used to describe both time and position dependence of
3411:
1173:
113:
5883:
2073:{\displaystyle \lambda _{D}={\frac {\mathrm {0.304nm} }{\sqrt {c_{0}}}}}
5872:
3415:
3394:
The linearized Poisson–Boltzmann equation can be used to calculate the
5886:
Adaptive Fast Multipole Poisson–Boltzmann Solver, free and open-source
5684:
5520:
3498:{\displaystyle \Delta G^{\text{el}}=\int ^{\tau }qU(\tau ')\,d\tau '}
5281:
426:
The freedom of movement of ions in solution can be accounted for by
5860:
4350:
3916:{\displaystyle \Delta G^{\text{ef}}=\int _{V}{\frac {p^{f}U}{2}}dV}
3751:
Note that the free energy is independent of the charging pathway .
3419:
1192:; however, with certain geometries, it can be solved analytically.
923:{\displaystyle c^{+}=c_{0}\cdot e^{\frac {-e\psi (x,y,z)}{k_{B}T}}}
1783:{\displaystyle e\left\vert \psi \right\vert \ll k_{\mathrm {B} }T}
824:{\displaystyle c^{-}=c_{0}\cdot e^{\frac {e\psi (x,y,z)}{k_{B}T}}}
92:
Thermal motion of ions accounted for; ions behave as point charges
2021:
water with a monovalent salt. The Debye length equation is then:
5423:
Department of Chemical Engineering, Carnegie Mellon University.
5875:
Matched Interface & Boundary based Poisson–Boltzmann solver
5666:
Gruziel, Magdalena; Grochowski, Pawel; Trylska, Joanna (2008).
3013:, substitute this for y in the previous equation and solve for
648:
5606:
1630:{\displaystyle R_{eD}={\sqrt {\frac {kT}{4\pi e^{2}n_{e0}}}}}
1142:{\displaystyle \rho _{e}=e{(c^{+}-c^{-})}=c_{0}e\cdot \left.}
5761:
3399:
5665:
1674:
5369:
5267:
4855:
4760:
4714:
4400:
4275:
4129:
4027:
3931:
3858:
3760:
3542:
3511:
3436:
3291:
3041:
3019:
2969:
2743:
2509:
2472:
2290:
2234:
2117:
2088:
2029:
1997:
1950:
1863:
1834:
1796:
1747:
1689:
1643:
1565:
1522:
1404:
1209:
947:
837:
741:
702:
666:
633:
598:
569:
535:
440:
434:
is used to calculate the local ion density such that
391:
358:
329:
140:
3398:
and free energy of highly charged molecules such as
5394:
4021:Entropic free energy of mixing of mobile species =
5096:
4839:
4743:
4697:
4339:
4258:
4113:
4011:
3915:
3840:
3741:
3520:
3497:
3342:{\displaystyle y_{0}={\frac {e\psi _{0}}{k_{B}T}}}
3341:
3276:
3025:
3005:
2953:
2725:
2493:
2456:
2270:
2208:
2101:
2072:
2013:
1981:
1935:
1847:
1816:
1782:
1726:
1659:
1629:
1551:
1506:
1388:
1141:
922:
823:
727:
688:
639:
613:
582:
553:
518:
404:
371:
342:
312:
5715:
1727:{\displaystyle \psi =\psi _{0}e^{-\mathrm {K} x}}
1164:as Gouy-Chapman theory; in solution chemistry as
5901:
5579:Cell Physiology Sourcebook: A Molecular Approach
3006:{\displaystyle y\equiv {\frac {e\psi }{k_{B}T}}}
2271:{\displaystyle y\equiv {\frac {e\psi }{k_{B}T}}}
1736:
123:A simple planar case for the Gouy–Chapman model
5346:(2nd ed.). Weinheim, Germany: Wiley-VCH.
4381:and self-consistent average potential for the
1982:{\displaystyle \lambda _{D}=\mathrm {K} ^{-1}}
1174:Derjaguin–Landau–Verwey–Overbeek (DLVO) theory
5341:
3385:
127:The Poisson–Boltzmann equation describes the
4123:Entropic free energy of mixing of solvent =
3425:
1184:Because the Poisson–Boltzmann equation is a
350:is the local electric charge density in C/m,
36:
5869:– A Poisson–Boltzmann electrostatics solver
5804:
5365:
5363:
5337:
5335:
5333:
5331:
5329:
5327:
1188:of the second order, it is commonly solved
5576:
5325:
5323:
5321:
5319:
5317:
5315:
5313:
5311:
5309:
5307:
5176:
1552:{\displaystyle \psi ={\frac {e\Phi }{kT}}}
5818:
5741:
5692:
5642:
5498:
5190:
3483:
1817:{\displaystyle \psi \leq \mathrm {25mV} }
5544:Journal of Colloid and Interface Science
5492:
5360:
4744:{\displaystyle \rho \rho _{1}+\rho _{2}}
3363:
118:
5541:
5535:
5418:
5416:
5304:
3372:
1179:
49:in 1910 and 1913, respectively. In the
5902:
5668:"The Poisson-Boltzmann model for tRNA"
5342:Butt, H.; Graf, L.; Kappl, M. (2006).
5237:
3033:. The following equation is rendered.
2218:
5757:
5755:
5753:
5602:
5600:
5598:
5390:
5388:
1675:Low-potential vs high-potential cases
561:is the ion concentration at the bulk,
5470:
5468:
5413:
5263:
5261:
4364:
112:The Gouy–Chapman model explains the
5344:Physics and Chemistry of Interfaces
3414:. This takes into account both the
1741:Strictly, low potential means that
1154:
934:is the charge of an electron, 1.602
13:
5750:
5595:
5474:
5385:
4675:
4660:
4613:
4487:
4472:
4324:
4308:
4292:
4276:
4182:
4130:
4098:
4028:
3932:
3859:
3825:
3809:
3793:
3777:
3761:
3600:
3543:
3528:is the final charge on the sphere
3437:
3262:
3168:
2939:
2845:
2711:
2617:
2300:
2160:
2119:
2052:
2049:
1966:
1920:
1865:
1810:
1807:
1771:
1715:
1535:
1516:It uses a dimensionless potential
1368:
1319:
605:
500:
412:is the permittivity of free space,
249:
235:
212:
198:
175:
161:
142:
14:
5931:
5861:Adaptive Poisson–Boltzmann Solver
5854:
5581:(3rd ed.). San Diego: Acad.
5465:
5258:
3700:
614:{\displaystyle k_{\mathrm {B} }}
405:{\displaystyle \varepsilon _{0}}
372:{\displaystyle \varepsilon _{r}}
5798:
5743:10.1590/S0103-97332000000200023
5709:
5659:
5570:
5475:Lu, B. Z.; et al. (2008).
5244:. Academic Press. p. 318.
5179:The European Physical Journal E
5121:
3925:Electrostatic mobile charges =
5910:Eponymous equations of physics
5441:
5231:
5170:
4907:
4888:
4708:the system can be found using
4651:
4648:
4631:
4619:
3852:Electrostatic fixed charges =
3708:
3696:
3480:
3469:
3247:
3213:
3153:
3119:
2924:
2890:
2830:
2796:
2696:
2662:
2602:
2568:
2448:
2414:
2411:
2384:
2379:
2345:
2342:
2315:
1357:
1351:
1308:
1302:
1109:
1091:
1053:
1035:
991:
965:
898:
880:
799:
781:
1:
5635:10.1016/S0006-3495(99)77173-0
5564:10.1016/S0021-9797(02)00142-X
5370:New Mexico State University.
5164:
3351:
1737:Low-potential case conditions
1195:
728:{\displaystyle W^{-}=-e\psi }
5722:Brazilian Journal of Physics
2014:{\displaystyle 25^{\circ }C}
1848:{\displaystyle \lambda _{D}}
689:{\displaystyle W^{+}=e\psi }
379:is the dielectric constant (
7:
5238:Attard, Phil (2002-08-07).
5152:
3391:electrostatic free energy.
10:
5936:
5784:10.1088/1674-1056/18/7/059
5397:"Chemistry 465 Lecture 10"
3386:Physiological applications
31:
21:Poisson–Boltzmann equation
5837:10.1209/epl/i2000-00495-1
5501:Technical Physics Letters
5395:Simon Fraser University.
3426:Electrostatic free energy
2466:This equation solved for
554:{\displaystyle c_{i}^{0}}
343:{\displaystyle \rho _{e}}
129:electrochemical potential
70:Important characteristics
37:Background and derivation
2280:trigonometric identities
5879:CHARMM-GUI: PBEQ Solver
5577:Sperelakis, N. (2012).
5449:"Electric Double Layer"
5372:"Electric Double Layer"
3396:electrostatic potential
2494:{\displaystyle e^{y/2}}
5098:
4841:
4745:
4699:
4341:
4260:
4115:
4013:
3917:
3842:
3743:
3522:
3521:{\displaystyle \tau q}
3499:
3369:
3343:
3278:
3027:
3007:
2955:
2727:
2495:
2458:
2272:
2210:
2103:
2074:
2015:
1983:
1937:
1849:
1818:
1784:
1728:
1661:
1660:{\displaystyle n_{e0}}
1631:
1553:
1508:
1390:
1143:
924:
825:
729:
690:
660:can be represented by
647:is the temperature in
641:
615:
584:
555:
520:
406:
373:
344:
314:
124:
5482:Commun. Comput. Phys.
5209:10.1007/s101890050023
5099:
4842:
4746:
4700:
4342:
4261:
4116:
4014:
3918:
3843:
3744:
3523:
3500:
3367:
3344:
3279:
3028:
3026:{\displaystyle \psi }
3008:
2956:
2728:
2496:
2459:
2273:
2211:
2104:
2102:{\displaystyle c_{0}}
2075:
2016:
1984:
1938:
1850:
1819:
1785:
1729:
1662:
1632:
1554:
1509:
1391:
1144:
925:
826:
730:
691:
642:
616:
585:
583:{\displaystyle W_{i}}
556:
521:
407:
381:relative permittivity
374:
345:
315:
122:
47:David Leonard Chapman
4853:
4758:
4712:
4398:
4273:
4127:
4025:
3929:
3856:
3758:
3540:
3509:
3434:
3408:phospholipid bilayer
3373:General applications
3289:
3039:
3017:
2967:
2741:
2507:
2470:
2288:
2232:
2115:
2086:
2027:
1995:
1948:
1861:
1832:
1794:
1745:
1687:
1641:
1563:
1520:
1402:
1207:
1186:partial differential
1180:Solving analytically
945:
835:
739:
700:
664:
631:
596:
567:
533:
438:
428:Boltzmann statistics
389:
356:
327:
138:
5920:Colloidal chemistry
5829:2000EL.....52..705M
5807:Europhysics Letters
5776:2009ChPhB..18.2975Z
5734:2000BrJPh..30..403C
5627:1999BpJ....76....1F
5614:Biophysical Journal
5556:2003JCIS..258...45T
5513:2005TePhL..31..204D
5201:2000EPJE....1..203N
4563:
4375:dissipative systems
4186:
4102:
3604:
3378:further insight on
2219:High-potential case
1166:Debye–Huckel theory
550:
468:
5915:Molecular dynamics
5894:Philip T. Gressman
5094:
4837:
4741:
4695:
4549:
4391:electronic density
4379:Boltzmann equation
4337:
4256:
4172:
4171:
4111:
4088:
4009:
3970:
3913:
3838:
3739:
3590:
3589:
3518:
3495:
3370:
3339:
3274:
3023:
3003:
2951:
2723:
2491:
2454:
2268:
2206:
2099:
2070:
2011:
1979:
1933:
1845:
1814:
1780:
1724:
1657:
1627:
1549:
1504:
1386:
1139:
920:
821:
725:
686:
637:
623:Boltzmann constant
611:
580:
551:
536:
516:
454:
432:Boltzmann equation
420:electric potential
402:
369:
340:
310:
125:
51:Gouy-Chapman model
43:Louis Georges Gouy
5764:Chinese Physics B
5685:10.1002/jcc.20953
5679:(12): 1970–1981.
5588:978-0-12-387738-3
5521:10.1134/1.1894433
5353:978-3-527-40629-6
5251:978-0-12-066321-7
5085:
5084:
5004:
4968:
4935:
4883:
4809:
4682:
4609:
4572:
4538:
4501:
4467:
4365:Materials science
4359:colloid chemistry
4334:
4318:
4302:
4286:
4239:
4162:
4140:
4103:
4038:
4001:
3961:
3942:
3905:
3869:
3835:
3819:
3803:
3787:
3771:
3726:
3657:
3580:
3553:
3447:
3337:
3272:
3071:
3001:
2949:
2721:
2452:
2266:
2204:
2170:
2068:
2067:
1931:
1930:
1625:
1624:
1547:
1470:
1450:
1437:
1378:
1329:
1279:
1242:
1170:colloid chemistry
1128:
1072:
917:
818:
640:{\displaystyle T}
510:
383:) of the solvent,
305:
263:
226:
189:
110:
109:
59:Hermann Helmholtz
5927:
5849:
5848:
5822:
5820:cond-mat/0009376
5802:
5796:
5795:
5770:(2): 2975–2980.
5759:
5748:
5747:
5745:
5713:
5707:
5706:
5696:
5673:J. Comput. Chem.
5663:
5657:
5656:
5646:
5604:
5593:
5592:
5574:
5568:
5567:
5539:
5533:
5532:
5496:
5490:
5489:
5472:
5463:
5462:
5460:
5459:
5445:
5439:
5438:
5436:
5434:
5429:
5420:
5411:
5410:
5408:
5406:
5401:
5392:
5383:
5382:
5380:
5378:
5367:
5358:
5357:
5339:
5302:
5301:
5270:J. Mol. Recognit
5265:
5256:
5255:
5235:
5229:
5228:
5194:
5192:cond-mat/9902085
5174:
5134:
5133:
5129:
5109:
5103:
5101:
5100:
5095:
5093:
5089:
5088:
5087:
5086:
5083:
5082:
5081:
5072:
5071:
5058:
5057:
5056:
5047:
5046:
5030:
5029:
5005:
5003:
5002:
5001:
4989:
4988:
4979:
4978:
4969:
4964:
4958:
4957:
4956:
4952:
4951:
4936:
4934:
4933:
4924:
4916:
4906:
4905:
4884:
4879:
4870:
4865:
4864:
4846:
4844:
4843:
4838:
4836:
4835:
4831:
4830:
4810:
4808:
4807:
4806:
4790:
4789:
4788:
4775:
4770:
4769:
4750:
4748:
4747:
4742:
4740:
4739:
4727:
4726:
4704:
4702:
4701:
4696:
4694:
4683:
4681:
4673:
4672:
4671:
4658:
4647:
4612:
4611:
4610:
4608:
4607:
4598:
4597:
4582:
4573:
4565:
4562:
4557:
4545:
4541:
4540:
4539:
4537:
4536:
4527:
4519:
4502:
4500:
4499:
4498:
4485:
4484:
4483:
4470:
4468:
4463:
4462:
4461:
4452:
4451:
4438:
4433:
4432:
4420:
4419:
4410:
4409:
4387:electric current
4383:Poisson equation
4346:
4344:
4343:
4338:
4336:
4335:
4332:
4320:
4319:
4316:
4304:
4303:
4300:
4288:
4287:
4284:
4265:
4263:
4262:
4257:
4249:
4245:
4244:
4240:
4238:
4230:
4223:
4222:
4209:
4185:
4180:
4170:
4161:
4160:
4142:
4141:
4138:
4120:
4118:
4117:
4112:
4104:
4101:
4096:
4087:
4086:
4077:
4069:
4068:
4059:
4058:
4040:
4039:
4036:
4018:
4016:
4015:
4010:
4002:
3997:
3990:
3989:
3980:
3979:
3969:
3959:
3957:
3956:
3944:
3943:
3940:
3922:
3920:
3919:
3914:
3906:
3901:
3897:
3896:
3886:
3884:
3883:
3871:
3870:
3867:
3847:
3845:
3844:
3839:
3837:
3836:
3833:
3821:
3820:
3817:
3805:
3804:
3801:
3789:
3788:
3785:
3773:
3772:
3769:
3748:
3746:
3745:
3740:
3732:
3728:
3727:
3725:
3717:
3716:
3715:
3703:
3688:
3680:
3679:
3667:
3663:
3662:
3658:
3656:
3648:
3641:
3640:
3627:
3603:
3598:
3588:
3568:
3567:
3555:
3554:
3551:
3527:
3525:
3524:
3519:
3504:
3502:
3501:
3496:
3494:
3479:
3462:
3461:
3449:
3448:
3445:
3348:
3346:
3345:
3340:
3338:
3336:
3332:
3331:
3321:
3320:
3319:
3306:
3301:
3300:
3283:
3281:
3280:
3275:
3273:
3271:
3270:
3269:
3265:
3240:
3239:
3235:
3230:
3229:
3203:
3202:
3198:
3193:
3192:
3177:
3176:
3175:
3171:
3146:
3145:
3141:
3136:
3135:
3109:
3108:
3104:
3099:
3098:
3083:
3072:
3067:
3063:
3062:
3049:
3032:
3030:
3029:
3024:
3012:
3010:
3009:
3004:
3002:
3000:
2996:
2995:
2985:
2977:
2960:
2958:
2957:
2952:
2950:
2948:
2947:
2946:
2942:
2917:
2916:
2912:
2907:
2906:
2880:
2879:
2875:
2870:
2869:
2854:
2853:
2852:
2848:
2823:
2822:
2818:
2813:
2812:
2786:
2785:
2781:
2776:
2775:
2760:
2732:
2730:
2729:
2724:
2722:
2720:
2719:
2718:
2714:
2689:
2688:
2684:
2679:
2678:
2652:
2651:
2647:
2642:
2641:
2626:
2625:
2624:
2620:
2595:
2594:
2590:
2585:
2584:
2558:
2557:
2553:
2548:
2547:
2532:
2527:
2526:
2522:
2501:is shown below.
2500:
2498:
2497:
2492:
2490:
2489:
2485:
2463:
2461:
2460:
2455:
2453:
2451:
2441:
2440:
2436:
2431:
2430:
2404:
2403:
2399:
2382:
2372:
2371:
2367:
2362:
2361:
2335:
2334:
2330:
2313:
2308:
2307:
2303:
2277:
2275:
2274:
2269:
2267:
2265:
2261:
2260:
2250:
2242:
2215:
2213:
2212:
2207:
2205:
2203:
2202:
2197:
2196:
2195:
2184:
2183:
2171:
2169:
2165:
2164:
2163:
2153:
2152:
2139:
2138:
2129:
2127:
2122:
2108:
2106:
2105:
2100:
2098:
2097:
2079:
2077:
2076:
2071:
2069:
2066:
2065:
2056:
2055:
2044:
2039:
2038:
2020:
2018:
2017:
2012:
2007:
2006:
1988:
1986:
1985:
1980:
1978:
1977:
1969:
1960:
1959:
1942:
1940:
1939:
1934:
1932:
1929:
1925:
1924:
1923:
1913:
1912:
1899:
1898:
1897:
1888:
1887:
1874:
1873:
1868:
1854:
1852:
1851:
1846:
1844:
1843:
1823:
1821:
1820:
1815:
1813:
1789:
1787:
1786:
1781:
1776:
1775:
1774:
1761:
1733:
1731:
1730:
1725:
1723:
1722:
1718:
1705:
1704:
1666:
1664:
1663:
1658:
1656:
1655:
1636:
1634:
1633:
1628:
1626:
1623:
1622:
1621:
1609:
1608:
1592:
1584:
1583:
1578:
1577:
1558:
1556:
1555:
1550:
1548:
1546:
1538:
1530:
1513:
1511:
1510:
1505:
1503:
1502:
1484:
1483:
1471:
1469:
1461:
1453:
1451:
1443:
1438:
1436:
1435:
1434:
1421:
1417:
1416:
1406:
1395:
1393:
1392:
1387:
1385:
1381:
1380:
1379:
1377:
1373:
1372:
1371:
1360:
1340:
1331:
1330:
1328:
1324:
1323:
1322:
1311:
1294:
1280:
1278:
1277:
1276:
1263:
1259:
1258:
1248:
1243:
1241:
1240:
1239:
1226:
1222:
1221:
1211:
1162:electrochemistry
1155:Related theories
1148:
1146:
1145:
1140:
1135:
1131:
1130:
1129:
1127:
1123:
1122:
1112:
1083:
1074:
1073:
1071:
1067:
1066:
1056:
1024:
1007:
1006:
994:
990:
989:
977:
976:
957:
956:
937:
929:
927:
926:
921:
919:
918:
916:
912:
911:
901:
869:
860:
859:
847:
846:
830:
828:
827:
822:
820:
819:
817:
813:
812:
802:
773:
764:
763:
751:
750:
734:
732:
731:
726:
712:
711:
695:
693:
692:
687:
676:
675:
659:
646:
644:
643:
638:
620:
618:
617:
612:
610:
609:
608:
589:
587:
586:
581:
579:
578:
560:
558:
557:
552:
549:
544:
525:
523:
522:
517:
512:
511:
509:
505:
504:
503:
492:
491:
490:
477:
467:
462:
450:
449:
417:
411:
409:
408:
403:
401:
400:
378:
376:
375:
370:
368:
367:
349:
347:
346:
341:
339:
338:
319:
317:
316:
311:
306:
304:
303:
302:
293:
292:
282:
281:
272:
264:
262:
261:
260:
247:
243:
242:
232:
227:
225:
224:
223:
210:
206:
205:
195:
190:
188:
187:
186:
173:
169:
168:
158:
150:
149:
133:Poisson equation
64:
63:
5935:
5934:
5930:
5929:
5928:
5926:
5925:
5924:
5900:
5899:
5857:
5852:
5803:
5799:
5760:
5751:
5714:
5710:
5664:
5660:
5605:
5596:
5589:
5575:
5571:
5540:
5536:
5497:
5493:
5488:(5): 973–1009 .
5473:
5466:
5457:
5455:
5447:
5446:
5442:
5432:
5430:
5427:
5421:
5414:
5404:
5402:
5399:
5393:
5386:
5376:
5374:
5368:
5361:
5354:
5340:
5305:
5282:10.1002/jmr.577
5266:
5259:
5252:
5236:
5232:
5175:
5171:
5167:
5155:
5135:
5131:
5127:
5125:
5124:
5107:
5077:
5073:
5067:
5063:
5059:
5052:
5048:
5042:
5038:
5031:
5028:
5021:
5017:
5010:
5006:
4994:
4990:
4984:
4980:
4974:
4970:
4963:
4959:
4944:
4940:
4929:
4925:
4917:
4915:
4914:
4910:
4898:
4894:
4878:
4871:
4869:
4860:
4856:
4854:
4851:
4850:
4823:
4819:
4815:
4811:
4799:
4795:
4791:
4784:
4780:
4776:
4774:
4765:
4761:
4759:
4756:
4755:
4735:
4731:
4722:
4718:
4713:
4710:
4709:
4687:
4674:
4667:
4663:
4659:
4657:
4640:
4603:
4599:
4590:
4583:
4581:
4578:
4577:
4564:
4558:
4553:
4532:
4528:
4520:
4518:
4514:
4507:
4503:
4494:
4490:
4486:
4479:
4475:
4471:
4469:
4457:
4453:
4447:
4443:
4439:
4437:
4428:
4424:
4415:
4411:
4405:
4401:
4399:
4396:
4395:
4367:
4331:
4327:
4315:
4311:
4299:
4295:
4283:
4279:
4274:
4271:
4270:
4231:
4218:
4214:
4210:
4208:
4204:
4191:
4187:
4181:
4176:
4166:
4156:
4152:
4137:
4133:
4128:
4125:
4124:
4097:
4092:
4082:
4078:
4076:
4064:
4060:
4054:
4050:
4035:
4031:
4026:
4023:
4022:
3985:
3981:
3975:
3971:
3965:
3960:
3958:
3952:
3948:
3939:
3935:
3930:
3927:
3926:
3892:
3888:
3887:
3885:
3879:
3875:
3866:
3862:
3857:
3854:
3853:
3832:
3828:
3816:
3812:
3800:
3796:
3784:
3780:
3768:
3764:
3759:
3756:
3755:
3718:
3711:
3707:
3699:
3689:
3687:
3675:
3671:
3649:
3636:
3632:
3628:
3626:
3622:
3609:
3605:
3599:
3594:
3584:
3573:
3569:
3563:
3559:
3550:
3546:
3541:
3538:
3537:
3510:
3507:
3506:
3487:
3472:
3457:
3453:
3444:
3440:
3435:
3432:
3431:
3428:
3388:
3375:
3354:
3327:
3323:
3322:
3315:
3311:
3307:
3305:
3296:
3292:
3290:
3287:
3286:
3261:
3257:
3253:
3231:
3225:
3221:
3220:
3216:
3194:
3188:
3184:
3183:
3179:
3178:
3167:
3163:
3159:
3137:
3131:
3127:
3126:
3122:
3100:
3094:
3090:
3089:
3085:
3084:
3082:
3058:
3054:
3050:
3048:
3040:
3037:
3036:
3018:
3015:
3014:
2991:
2987:
2986:
2978:
2976:
2968:
2965:
2964:
2938:
2934:
2930:
2908:
2902:
2898:
2897:
2893:
2871:
2865:
2861:
2860:
2856:
2855:
2844:
2840:
2836:
2814:
2808:
2804:
2803:
2799:
2777:
2771:
2767:
2766:
2762:
2761:
2759:
2742:
2739:
2738:
2710:
2706:
2702:
2680:
2674:
2670:
2669:
2665:
2643:
2637:
2633:
2632:
2628:
2627:
2616:
2612:
2608:
2586:
2580:
2576:
2575:
2571:
2549:
2543:
2539:
2538:
2534:
2533:
2531:
2518:
2514:
2510:
2508:
2505:
2504:
2481:
2477:
2473:
2471:
2468:
2467:
2432:
2426:
2422:
2421:
2417:
2395:
2391:
2387:
2383:
2363:
2357:
2353:
2352:
2348:
2326:
2322:
2318:
2314:
2312:
2299:
2295:
2291:
2289:
2286:
2285:
2256:
2252:
2251:
2243:
2241:
2233:
2230:
2229:
2221:
2198:
2191:
2187:
2186:
2185:
2179:
2175:
2159:
2158:
2154:
2148:
2144:
2140:
2134:
2130:
2128:
2126:
2118:
2116:
2113:
2112:
2093:
2089:
2087:
2084:
2083:
2061:
2057:
2045:
2043:
2034:
2030:
2028:
2025:
2024:
2002:
1998:
1996:
1993:
1992:
1970:
1965:
1964:
1955:
1951:
1949:
1946:
1945:
1919:
1918:
1914:
1908:
1904:
1900:
1893:
1889:
1883:
1879:
1875:
1872:
1864:
1862:
1859:
1858:
1839:
1835:
1833:
1830:
1829:
1803:
1795:
1792:
1791:
1770:
1769:
1765:
1751:
1746:
1743:
1742:
1739:
1714:
1710:
1706:
1700:
1696:
1688:
1685:
1684:
1677:
1648:
1644:
1642:
1639:
1638:
1614:
1610:
1604:
1600:
1593:
1585:
1582:
1570:
1566:
1564:
1561:
1560:
1539:
1531:
1529:
1521:
1518:
1517:
1495:
1491:
1479:
1475:
1462:
1454:
1452:
1442:
1430:
1426:
1422:
1412:
1408:
1407:
1405:
1403:
1400:
1399:
1367:
1366:
1362:
1361:
1341:
1339:
1335:
1318:
1317:
1313:
1312:
1295:
1293:
1289:
1288:
1284:
1272:
1268:
1264:
1254:
1250:
1249:
1247:
1235:
1231:
1227:
1217:
1213:
1212:
1210:
1208:
1205:
1204:
1198:
1182:
1157:
1118:
1114:
1113:
1084:
1082:
1078:
1062:
1058:
1057:
1025:
1023:
1019:
1018:
1014:
1002:
998:
985:
981:
972:
968:
964:
952:
948:
946:
943:
942:
935:
907:
903:
902:
870:
868:
864:
855:
851:
842:
838:
836:
833:
832:
808:
804:
803:
774:
772:
768:
759:
755:
746:
742:
740:
737:
736:
707:
703:
701:
698:
697:
671:
667:
665:
662:
661:
657:
632:
629:
628:
604:
603:
599:
597:
594:
593:
574:
570:
568:
565:
564:
545:
540:
534:
531:
530:
499:
498:
494:
493:
486:
482:
478:
476:
472:
463:
458:
445:
441:
439:
436:
435:
415:
396:
392:
390:
387:
386:
363:
359:
357:
354:
353:
334:
330:
328:
325:
324:
298:
294:
288:
284:
283:
277:
273:
271:
256:
252:
248:
238:
234:
233:
231:
219:
215:
211:
201:
197:
196:
194:
182:
178:
174:
164:
160:
159:
157:
145:
141:
139:
136:
135:
39:
34:
17:
12:
11:
5:
5933:
5923:
5922:
5917:
5912:
5898:
5897:
5887:
5881:
5876:
5870:
5864:
5856:
5855:External links
5853:
5851:
5850:
5813:(6): 705–711.
5797:
5749:
5728:(2): 403–409.
5708:
5658:
5594:
5587:
5569:
5534:
5507:(3): 204–207.
5491:
5464:
5440:
5412:
5384:
5359:
5352:
5303:
5276:(6): 379–385.
5257:
5250:
5230:
5185:(2): 203–214.
5168:
5166:
5163:
5162:
5161:
5154:
5151:
5123:
5120:
5110:is called the
5105:
5104:
5092:
5080:
5076:
5070:
5066:
5062:
5055:
5051:
5045:
5041:
5037:
5034:
5027:
5024:
5020:
5016:
5013:
5009:
5000:
4997:
4993:
4987:
4983:
4977:
4973:
4967:
4962:
4955:
4950:
4947:
4943:
4939:
4932:
4928:
4923:
4920:
4913:
4909:
4904:
4901:
4897:
4893:
4890:
4887:
4882:
4877:
4874:
4868:
4863:
4859:
4848:
4834:
4829:
4826:
4822:
4818:
4814:
4805:
4802:
4798:
4794:
4787:
4783:
4779:
4773:
4768:
4764:
4738:
4734:
4730:
4725:
4721:
4717:
4693:
4690:
4686:
4680:
4677:
4670:
4666:
4662:
4656:
4653:
4650:
4646:
4643:
4639:
4636:
4633:
4630:
4627:
4624:
4621:
4618:
4615:
4606:
4602:
4596:
4593:
4589:
4586:
4580:
4576:
4571:
4568:
4561:
4556:
4552:
4548:
4544:
4535:
4531:
4526:
4523:
4517:
4513:
4510:
4506:
4497:
4493:
4489:
4482:
4478:
4474:
4466:
4460:
4456:
4450:
4446:
4442:
4436:
4431:
4427:
4423:
4418:
4414:
4408:
4404:
4366:
4363:
4330:
4326:
4323:
4314:
4310:
4307:
4298:
4294:
4291:
4282:
4278:
4267:
4266:
4255:
4252:
4248:
4243:
4237:
4234:
4229:
4226:
4221:
4217:
4213:
4207:
4203:
4200:
4197:
4194:
4190:
4184:
4179:
4175:
4169:
4165:
4159:
4155:
4151:
4148:
4145:
4136:
4132:
4121:
4110:
4107:
4100:
4095:
4091:
4085:
4081:
4075:
4072:
4067:
4063:
4057:
4053:
4049:
4046:
4043:
4034:
4030:
4019:
4008:
4005:
4000:
3996:
3993:
3988:
3984:
3978:
3974:
3968:
3964:
3955:
3951:
3947:
3938:
3934:
3923:
3912:
3909:
3904:
3900:
3895:
3891:
3882:
3878:
3874:
3865:
3861:
3831:
3827:
3824:
3815:
3811:
3808:
3799:
3795:
3792:
3783:
3779:
3776:
3767:
3763:
3738:
3735:
3731:
3724:
3721:
3714:
3710:
3706:
3702:
3698:
3695:
3692:
3686:
3683:
3678:
3674:
3670:
3666:
3661:
3655:
3652:
3647:
3644:
3639:
3635:
3631:
3625:
3621:
3618:
3615:
3612:
3608:
3602:
3597:
3593:
3587:
3583:
3579:
3576:
3572:
3566:
3562:
3558:
3549:
3545:
3534:Euler-Lagrange
3517:
3514:
3493:
3490:
3486:
3482:
3478:
3475:
3471:
3468:
3465:
3460:
3456:
3452:
3443:
3439:
3427:
3424:
3387:
3384:
3380:electrostatics
3374:
3371:
3353:
3350:
3335:
3330:
3326:
3318:
3314:
3310:
3304:
3299:
3295:
3268:
3264:
3260:
3256:
3252:
3249:
3246:
3243:
3238:
3234:
3228:
3224:
3219:
3215:
3212:
3209:
3206:
3201:
3197:
3191:
3187:
3182:
3174:
3170:
3166:
3162:
3158:
3155:
3152:
3149:
3144:
3140:
3134:
3130:
3125:
3121:
3118:
3115:
3112:
3107:
3103:
3097:
3093:
3088:
3081:
3078:
3075:
3070:
3066:
3061:
3057:
3053:
3047:
3044:
3022:
2999:
2994:
2990:
2984:
2981:
2975:
2972:
2945:
2941:
2937:
2933:
2929:
2926:
2923:
2920:
2915:
2911:
2905:
2901:
2896:
2892:
2889:
2886:
2883:
2878:
2874:
2868:
2864:
2859:
2851:
2847:
2843:
2839:
2835:
2832:
2829:
2826:
2821:
2817:
2811:
2807:
2802:
2798:
2795:
2792:
2789:
2784:
2780:
2774:
2770:
2765:
2758:
2755:
2752:
2749:
2746:
2717:
2713:
2709:
2705:
2701:
2698:
2695:
2692:
2687:
2683:
2677:
2673:
2668:
2664:
2661:
2658:
2655:
2650:
2646:
2640:
2636:
2631:
2623:
2619:
2615:
2611:
2607:
2604:
2601:
2598:
2593:
2589:
2583:
2579:
2574:
2570:
2567:
2564:
2561:
2556:
2552:
2546:
2542:
2537:
2530:
2525:
2521:
2517:
2513:
2488:
2484:
2480:
2476:
2450:
2447:
2444:
2439:
2435:
2429:
2425:
2420:
2416:
2413:
2410:
2407:
2402:
2398:
2394:
2390:
2386:
2381:
2378:
2375:
2370:
2366:
2360:
2356:
2351:
2347:
2344:
2341:
2338:
2333:
2329:
2325:
2321:
2317:
2311:
2306:
2302:
2298:
2294:
2264:
2259:
2255:
2249:
2246:
2240:
2237:
2220:
2217:
2201:
2194:
2190:
2182:
2178:
2174:
2168:
2162:
2157:
2151:
2147:
2143:
2137:
2133:
2125:
2121:
2096:
2092:
2064:
2060:
2054:
2051:
2048:
2042:
2037:
2033:
2010:
2005:
2001:
1976:
1973:
1968:
1963:
1958:
1954:
1928:
1922:
1917:
1911:
1907:
1903:
1896:
1892:
1886:
1882:
1878:
1871:
1867:
1842:
1838:
1812:
1809:
1806:
1802:
1799:
1779:
1773:
1768:
1764:
1760:
1757:
1754:
1750:
1738:
1735:
1721:
1717:
1713:
1709:
1703:
1699:
1695:
1692:
1676:
1673:
1669:number density
1654:
1651:
1647:
1620:
1617:
1613:
1607:
1603:
1599:
1596:
1591:
1588:
1581:
1576:
1573:
1569:
1545:
1542:
1537:
1534:
1528:
1525:
1501:
1498:
1494:
1490:
1487:
1482:
1478:
1474:
1468:
1465:
1460:
1457:
1449:
1446:
1441:
1433:
1429:
1425:
1420:
1415:
1411:
1384:
1376:
1370:
1365:
1359:
1356:
1353:
1350:
1347:
1344:
1338:
1334:
1327:
1321:
1316:
1310:
1307:
1304:
1301:
1298:
1292:
1287:
1283:
1275:
1271:
1267:
1262:
1257:
1253:
1246:
1238:
1234:
1230:
1225:
1220:
1216:
1197:
1194:
1181:
1178:
1156:
1153:
1138:
1134:
1126:
1121:
1117:
1111:
1108:
1105:
1102:
1099:
1096:
1093:
1090:
1087:
1081:
1077:
1070:
1065:
1061:
1055:
1052:
1049:
1046:
1043:
1040:
1037:
1034:
1031:
1028:
1022:
1017:
1013:
1010:
1005:
1001:
997:
993:
988:
984:
980:
975:
971:
967:
963:
960:
955:
951:
915:
910:
906:
900:
897:
894:
891:
888:
885:
882:
879:
876:
873:
867:
863:
858:
854:
850:
845:
841:
816:
811:
807:
801:
798:
795:
792:
789:
786:
783:
780:
777:
771:
767:
762:
758:
754:
749:
745:
724:
721:
718:
715:
710:
706:
685:
682:
679:
674:
670:
653:
652:
636:
626:
607:
602:
591:
577:
573:
562:
548:
543:
539:
515:
508:
502:
497:
489:
485:
481:
475:
471:
466:
461:
457:
453:
448:
444:
424:
423:
413:
399:
395:
384:
366:
362:
351:
337:
333:
309:
301:
297:
291:
287:
280:
276:
270:
267:
259:
255:
251:
246:
241:
237:
230:
222:
218:
214:
209:
204:
200:
193:
185:
181:
177:
172:
167:
163:
156:
153:
148:
144:
108:
107:
104:
101:
97:
96:
93:
90:
86:
85:
82:
79:
75:
74:
71:
68:
38:
35:
33:
30:
15:
9:
6:
4:
3:
2:
5932:
5921:
5918:
5916:
5913:
5911:
5908:
5907:
5905:
5895:
5891:
5888:
5885:
5882:
5880:
5877:
5874:
5871:
5868:
5865:
5862:
5859:
5858:
5846:
5842:
5838:
5834:
5830:
5826:
5821:
5816:
5812:
5808:
5801:
5793:
5789:
5785:
5781:
5777:
5773:
5769:
5765:
5758:
5756:
5754:
5744:
5739:
5735:
5731:
5727:
5723:
5719:
5712:
5704:
5700:
5695:
5690:
5686:
5682:
5678:
5675:
5674:
5669:
5662:
5654:
5650:
5645:
5640:
5636:
5632:
5628:
5624:
5620:
5616:
5615:
5610:
5603:
5601:
5599:
5590:
5584:
5580:
5573:
5565:
5561:
5557:
5553:
5549:
5545:
5538:
5530:
5526:
5522:
5518:
5514:
5510:
5506:
5502:
5495:
5487:
5484:
5483:
5478:
5471:
5469:
5454:
5450:
5444:
5426:
5419:
5417:
5398:
5391:
5389:
5373:
5366:
5364:
5355:
5349:
5345:
5338:
5336:
5334:
5332:
5330:
5328:
5326:
5324:
5322:
5320:
5318:
5316:
5314:
5312:
5310:
5308:
5299:
5295:
5291:
5287:
5283:
5279:
5275:
5271:
5264:
5262:
5253:
5247:
5243:
5242:
5234:
5226:
5222:
5218:
5214:
5210:
5206:
5202:
5198:
5193:
5188:
5184:
5180:
5173:
5169:
5160:
5157:
5156:
5150:
5147:
5143:
5141:
5130:
5119:
5115:
5113:
5090:
5078:
5074:
5068:
5064:
5060:
5053:
5049:
5043:
5039:
5035:
5032:
5025:
5022:
5018:
5014:
5011:
5007:
4998:
4995:
4991:
4985:
4981:
4975:
4971:
4965:
4960:
4953:
4948:
4945:
4941:
4937:
4930:
4926:
4921:
4918:
4911:
4902:
4899:
4895:
4891:
4885:
4880:
4875:
4872:
4866:
4861:
4857:
4849:
4832:
4827:
4824:
4820:
4816:
4812:
4803:
4800:
4796:
4792:
4785:
4781:
4777:
4771:
4766:
4762:
4754:
4753:
4752:
4736:
4732:
4728:
4723:
4719:
4715:
4705:
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4371:semiconductor
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4355:nucleic acids
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939:
938:10 coulombs.
933:
913:
908:
904:
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886:
883:
877:
874:
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865:
861:
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563:
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506:
495:
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479:
473:
469:
464:
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421:
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397:
393:
385:
382:
364:
360:
352:
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331:
323:
322:
321:
307:
299:
295:
289:
285:
278:
274:
268:
265:
257:
253:
244:
239:
228:
220:
216:
207:
202:
191:
183:
179:
170:
165:
154:
151:
146:
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130:
121:
117:
115:
105:
102:
99:
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5550:(1): 45–49.
5547:
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5500:
5494:
5485:
5480:
5456:. Retrieved
5453:web.nmsu.edu
5452:
5443:
5431:. Retrieved
5403:. Retrieved
5375:. Retrieved
5343:
5273:
5269:
5240:
5233:
5182:
5178:
5172:
5159:Double layer
5148:
5144:
5140:permittivity
5136:
5122:Limitations
5116:
5112:Debye length
5106:
4706:
4368:
4348:
4268:
3753:
3750:
3536:functional:
3530:
3429:
3405:
3393:
3389:
3376:
3359:Debye length
3355:
3285:
3035:
2962:
2737:
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2465:
2284:
2222:
2111:
2081:
2023:
1990:
1944:
1857:
1827:Debye length
1740:
1678:
1667:denotes the
1515:
1397:
1203:
1199:
1183:
1158:
1150:
940:
931:
654:
425:
126:
111:
89:Gouy-Chapman
73:Assumptions
55:double layer
40:
20:
18:
5621:(1): 1–16.
3412:erythrocyte
1190:numerically
114:capacitance
5904:Categories
5458:2018-06-01
5165:References
3416:glycocalyx
3352:Conditions
2228:parameter
1855:equation.
1196:Geometries
26:mean-field
5792:250813154
5529:120529487
5225:119468015
5217:1292-8941
5075:ε
5065:ε
5026:−
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4972:ε
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4881:π
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3242:−
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3021:ψ
2983:ψ
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2928:⋅
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2825:−
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2691:−
2660:−
2614:−
2606:⋅
2597:−
2443:−
2337:−
2297:−
2248:ψ
2239:≡
2173:∑
2146:ε
2142:ε
2032:λ
2004:∘
1972:−
1953:λ
1906:ε
1902:ε
1837:λ
1801:≤
1798:ψ
1763:≪
1756:ψ
1712:−
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1691:ψ
1681:potential
1598:π
1536:Φ
1524:ψ
1500:ψ
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394:ε
361:ε
332:ρ
296:ε
286:ε
275:ρ
269:−
250:∂
245:ψ
236:∂
213:∂
208:ψ
199:∂
176:∂
171:ψ
162:∂
152:ψ
143:∇
78:Helmholtz
5845:18058376
5703:18432617
5298:17184352
5290:12501158
5153:See also
4692:′
4645:′
4595:′
4351:proteins
3492:′
3477:′
3420:spectrin
930:, where
5825:Bibcode
5772:Bibcode
5730:Bibcode
5694:2599918
5653:9876118
5644:1302495
5623:Bibcode
5552:Bibcode
5509:Bibcode
5433:June 1,
5405:June 1,
5377:June 1,
5197:Bibcode
1637:(where
649:kelvins
621:is the
418:is the
32:Origins
5884:AFMPB
5873:MIBPB
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4751:where
3848:where
3505:where
3410:of an
2082:where
526:where
430:. The
320:where
67:Theory
5841:S2CID
5815:arXiv
5788:S2CID
5525:S2CID
5428:(PDF)
5400:(PDF)
5294:S2CID
5221:S2CID
5187:arXiv
2047:0.304
1168:; in
100:Stern
5699:PMID
5649:PMID
5583:ISBN
5435:2014
5407:2014
5379:2014
5348:ISBN
5286:PMID
5246:ISBN
5213:ISSN
5128:edit
4389:and
4333:solv
4139:solv
3834:solv
3418:and
3400:tRNA
831:and
696:and
45:and
19:The
5867:Zap
5833:doi
5780:doi
5738:doi
5689:PMC
5681:doi
5639:PMC
5631:doi
5560:doi
5548:258
5517:doi
5278:doi
5205:doi
4847:and
4317:mob
4285:out
4199:exp
4037:mob
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3233:/
3227:0
3223:y
3218:e
3214:(
3208:1
3205:+
3200:2
3196:/
3190:0
3186:y
3181:e
3173:x
3169:K
3161:e
3154:)
3151:1
3143:2
3139:/
3133:0
3129:y
3124:e
3120:(
3117:+
3114:1
3111:+
3106:2
3102:/
3096:0
3092:y
3087:e
3069:e
3065:T
3060:B
3056:k
3052:2
3046:=
2998:T
2993:B
2989:k
2980:e
2971:y
2944:x
2940:K
2932:e
2925:)
2922:1
2914:2
2910:/
2904:0
2900:y
2895:e
2891:(
2885:1
2882:+
2877:2
2873:/
2867:0
2863:y
2858:e
2850:x
2846:K
2838:e
2831:)
2828:1
2820:2
2816:/
2810:0
2806:y
2801:e
2797:(
2794:+
2791:1
2788:+
2783:2
2779:/
2773:0
2769:y
2764:e
2751:2
2748:=
2745:y
2716:x
2712:K
2704:e
2697:)
2694:1
2686:2
2682:/
2676:0
2672:y
2667:e
2663:(
2657:1
2654:+
2649:2
2645:/
2639:0
2635:y
2630:e
2622:x
2618:K
2610:e
2603:)
2600:1
2592:2
2588:/
2582:0
2578:y
2573:e
2569:(
2566:+
2563:1
2560:+
2555:2
2551:/
2545:0
2541:y
2536:e
2529:=
2524:2
2520:/
2516:y
2512:e
2487:2
2483:/
2479:y
2475:e
2449:)
2446:1
2438:2
2434:/
2428:0
2424:y
2419:e
2415:(
2412:)
2409:1
2406:+
2401:2
2397:/
2393:y
2389:e
2385:(
2380:)
2377:1
2374:+
2369:2
2365:/
2359:0
2355:y
2350:e
2346:(
2343:)
2340:1
2332:2
2328:/
2324:y
2320:e
2316:(
2310:=
2305:x
2301:K
2293:e
2263:T
2258:B
2254:k
2245:e
2236:y
2200:2
2193:i
2189:Z
2181:i
2177:c
2167:T
2161:B
2156:k
2150:0
2136:2
2132:e
2124:=
2120:K
2095:0
2091:c
2063:0
2059:c
2053:m
2050:n
2041:=
2036:D
2009:C
1975:1
1967:K
1962:=
1957:D
1927:T
1921:B
1916:k
1910:0
1895:2
1891:e
1885:0
1881:c
1877:2
1870:=
1866:K
1841:D
1811:V
1808:m
1778:T
1772:B
1767:k
1759:|
1753:|
1749:e
1720:x
1716:K
1708:e
1702:0
1694:=
1653:0
1650:e
1646:n
1619:0
1616:e
1612:n
1606:2
1602:e
1595:4
1590:T
1587:k
1580:=
1575:D
1572:e
1568:R
1544:T
1541:k
1533:e
1527:=
1493:e
1477:e
1473:=
1467:r
1464:d
1456:d
1448:r
1445:L
1440:+
1432:2
1428:r
1424:d
1414:2
1410:d
1383:]
1375:T
1369:B
1364:k
1358:)
1355:x
1352:(
1346:e
1337:e
1326:T
1320:B
1315:k
1309:)
1306:x
1303:(
1297:e
1291:e
1286:[
1274:0
1261:e
1256:0
1252:c
1245:=
1237:2
1233:x
1229:d
1219:2
1215:d
1137:.
1133:]
1125:T
1120:B
1116:k
1110:)
1107:z
1104:,
1101:y
1098:,
1095:x
1092:(
1086:e
1080:e
1069:T
1064:B
1060:k
1054:)
1051:z
1048:,
1045:y
1042:,
1039:x
1036:(
1030:e
1021:e
1016:[
1009:e
1004:0
1000:c
996:=
992:)
983:c
974:+
970:c
966:(
962:e
959:=
954:e
936:×
932:e
914:T
909:B
905:k
899:)
896:z
893:,
890:y
887:,
884:x
881:(
875:e
866:e
857:0
853:c
849:=
844:+
840:c
815:T
810:B
806:k
800:)
797:z
794:,
791:y
788:,
785:x
782:(
776:e
770:e
761:0
757:c
753:=
744:c
720:e
714:=
705:W
681:e
678:=
673:+
669:W
658:ψ
651:.
635:T
625:,
606:B
601:k
576:i
572:W
547:0
542:i
538:c
514:,
507:T
501:B
496:k
488:i
484:W
474:e
465:0
460:i
456:c
452:=
447:i
443:c
422:.
416:ψ
398:0
365:r
336:e
308:,
300:0
290:r
279:e
266:=
258:2
254:z
240:2
229:+
221:2
217:y
203:2
192:+
184:2
180:x
166:2
155:=
147:2
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