3376:
4714:
5113:
131:
329:
4396:. 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
4408:
5157:
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.
4863:
3372:, 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.
5149:
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
3293:
148:
3758:
3401:
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
1211:
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
4718:
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
1835:
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
5148:
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
3388:
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
5156:
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
4404:
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
666:
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
3367:
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
1694:. 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.
2742:
2970:
34:
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
4405:
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
3857:
2225:
4709:{\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'}
3413:
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.
5108:{\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)}
3049:
1523:
1952:
4356:
3550:
1405:
2234:
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
2473:
5128:
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.
4130:
127:-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.
2517:
4856:
2751:
3542:
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
4028:
4275:
324:{\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}}},}
3514:
1170:
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
535:
2089:
3932:
939:
3768:
1799:
840:
2125:
1646:
1158:
4388:
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
5153:
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.
1187:. 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.
3358:
1412:
1743:
3022:
2287:
1871:
4283:
1998:
1217:
2746:
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.
2298:
1568:
1833:
4760:
1409:
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:
4368:, 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
630:
421:
388:
2002:
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
2120:
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.
744:
2030:
1864:
705:
3288:{\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}}}}
5160:
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.
2293:
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.
570:
359:
3441:
The Poisson–Boltzmann equation can also be used to calculate the electrostatic free energy for hypothetically charging a sphere using the following charging integral:
2510:
3537:
1676:
3042:
2118:
599:
4035:
448:
3753:{\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}
656:
2037:
3433:
layers of the erythrocyte membrane. This information is useful for many reasons including the study of the mechanical stability of the erythrocyte membrane.
92:
Surface charge neutralized by a molecular layer of counter-ions; surface charge potential linearly dissipated from surface to counter-ions to satisfy charge
68:. 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
3444:
4768:
3379:
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.
3939:
749:
4137:
3375:
955:
5920:
3417:
Another example of utilizing the Poisson–Boltzmann equation is the determination of an electric potential profile at points perpendicular to the
5773:
Zhang Li-Zhi; Wang Zheng-Chuan (2009). "Analytical Solution to the Boltzmann-Poisson Equation and Its Application to MIS Tunneling Junctions".
5279:
Fogolari, F.; Brigo, A.; Molinari, H. (2002). "The Poisson–Boltzmann Equation for Biomolecular Electrostatics: a Tool for Structural Biology".
4280:
Finally, by combining the last three term the following equation representing the outer space contribution to the free energy density integral
2737:{\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}}}}
5597:
5362:
5260:
2965:{\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}}}}
1801:; however, the results that the equations yields are valid for a wider range of potentials, from 50–80mV. Nevertheless, at room temperature,
114:
Finite ion size and hydration sphere considered; some ions are specifically adsorbed by the surface in the plane, known as the Stern layer
3866:
845:
3299:
1755:
4380:
An analytical solution to the Poisson–Boltzmann equation can be used to describe an electron-electron interaction in a metal-insulator
5683:
5250:
1958:
1573:
952:
Substituting these Boltzmann relations into the local electric charge density expression, the following expression can be obtained
2290:
3765:
The above expression can be rewritten into separate free energy terms based on different contributions to the total free energy
39:
assumptions. From the Poisson–Boltzmann equation many other equations have been derived with a number of different assumptions.
4722:
5492:
61:
1682:
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.
5510:
D’Yachkov, L. G. (2005). "Analytical Solution of the Poisson–Boltzmann Equation in Cases of Spherical and Axial Symmetry".
3852:{\displaystyle \Delta G^{\text{el}}=\Delta G^{\text{ef}}+\Delta G^{\text{em}}+\Delta G^{\text{mob}}+\Delta G^{\text{solv}}}
2220:{\displaystyle \mathrm {K} ={\sqrt {{\frac {e^{2}}{\varepsilon \varepsilon _{0}k_{\mathrm {B} }T}}\sum c_{i}{Z_{i}}^{2}}}}
1196:
5930:
1697:
5925:
2977:
2242:
5553:
Tuinier, R. (2003). "Approximate Solutions to the Poisson–Boltzmann Equation in Spherical and Cylindrical Geometry".
64:, a charged solid comes into contact with an ionic solution, creating a layer of surface charges and counter-ions or
5169:
1162:
Finally the charge density can be substituted into the Poisson equation to produce the Poisson–Boltzmann equation.
65:
139:
5188:
Netz, R.R.; Orland, H. (2000-02-01). "Beyond Poisson-Boltzmann: Fluctuation effects and correlation functions".
1530:
95:
Thermal motion, ion diffusion, adsorption onto the surface, solvent/surface interactions considered negligible
1804:
1176:
72:
in the Helmholtz model. The Stern Layer model goes a step further and takes into account the finite ion size.
746:
respectively. These work equations can be substituted into the Boltzmann equation, producing two expressions
5436:"The Application of a Dynamic Stern Layer Model to Electrophoretic Mobility Measurements of Latex Particles"
3406:
1690:
When using the Poisson–Boltzmann equation, it is important to determine if the specific case is low or high
1518:{\displaystyle {\frac {d^{2}\psi }{dr^{2}}}+{\frac {L}{r}}{\frac {d\psi }{dr}}=e^{\psi }-\delta e^{-\psi }}
2236:
5907:
and Robert M. Strain, 2009, University of Pennsylvania, Department of Mathematics, Philadelphia, PA, USA.
1947:{\displaystyle \mathrm {K} ={\sqrt {\frac {2c_{0}e^{2}}{\varepsilon \varepsilon _{0}k_{\mathrm {B} }T}}}}
1212:
Poisson–Boltzmann equation solved analytically in terms of a second order derivative with respect to x.
391:
57:
4351:{\displaystyle \Delta G^{\text{out}}=\Delta G^{\text{em}}+\Delta G^{\text{mob}}+\Delta G^{\text{solv}}}
1400:{\displaystyle {\frac {d^{2}\psi }{dx^{2}}}={\frac {c_{0}e}{\varepsilon \varepsilon _{0}}}\cdot \left}
606:
399:
366:
5835:
5782:
5740:
5633:
5562:
5519:
5488:"Recent Progress in Numerical Methods for the Poisson-Boltzmann Equation in Biophysical Applications"
5459:
5207:
438:
5382:
5624:
2468:{\displaystyle e^{-\mathrm {K} x}={\frac {(e^{y/2}-1)(e^{y_{0}/2}+1)}{(e^{y/2}+1)(e^{y_{0}/2}-1)}}}
1570:
and the lengths are measured in units of the Debye electron radius in the region of zero potential
710:
142:
of ions in the diffuse layer. The three-dimensional potential distribution can be described by the
5874:– A free, open-source Poisson-Boltzmann electrostatics and biomolecular solvation software package
2005:
1842:
674:
5851:
5825:
5798:
5535:
5304:
5231:
5197:
4401:
4389:
4385:
1691:
1200:
633:
442:
430:
53:
543:
337:
5709:
5659:
5593:
5358:
5296:
5256:
5223:
4369:
1180:
69:
36:
5487:
2480:
5843:
5816:
Moreira, A. G.; Netz, R. R. (2000). "Strong-coupling theory for counter-ion distributions".
5790:
5748:
5699:
5691:
5649:
5641:
5570:
5527:
5435:
5288:
5215:
4397:
4393:
4125:{\displaystyle \Delta G^{\text{mob}}=kT\int _{V}c_{i}\ln {\frac {c_{i}}{c_{i}^{\infty }}}dV}
3519:
1651:
1172:
143:
3027:
2096:
577:
130:
5904:
601:
is the work required to move an ion closer to the surface from an infinitely far distance,
17:
5839:
5786:
5744:
5637:
5566:
5523:
5211:
106:
Finite ion size ignored; uniformly-charged surface; non-Coulombic interactions ignored
5704:
5678:
5654:
5619:
5407:
3544:
3390:
2289:, which is not to be confused with the spatial coordinate symbol, y. Employing several
1679:
641:
5645:
5618:
Fogolari, Federico; Zuccato, Pierfrancesco; Esposito, Gennaro; Viglino, Paola (1999).
5574:
5914:
5877:
5802:
5794:
5539:
5235:
4381:
4365:
3418:
667:
electric work to bring a charged cation or charged anion to a surface with potential
5855:
5753:
5728:
5308:
4851:{\displaystyle \rho _{1}\approx {\frac {aE_{z}}{2\lambda _{D1}}}e^{-\lambda _{D1}z}}
5900:
5150:
5122:
3369:
1837:
4023:{\displaystyle \Delta G^{\text{em}}=\int _{V}{\frac {\sum _{i}c_{i}z_{i}qU}{2}}dV}
117:
Stern layer is thin compared to particle size; fluid velocity = 0 in Stern layer
5901:
Global classical solutions of the Boltzmann equation with long-range interactions
4360:
These equations can act as simple geometry models for biological systems such as
4270:{\displaystyle \Delta G^{\text{solv}}=kT\int _{V}\sum _{i}c_{i}^{\infty }\leftdV}
5847:
3422:
1184:
124:
27:
Equation used for physiological interfaces, polymer science, and semiconductors
3426:
5889:
5729:"Solutions of non-linear Poisson–Boltzmann equation for erythrocyte membrane"
5252:
Thermodynamics and Statistical Mechanics: Equilibrium by Entropy Maximisation
5227:
5727:
Cruz, Frederico A. O.; Vilhena, Fernando S. D. S.; Cortez, Celia M. (2000).
5620:"Biomolecular Electrostatics with the Linearized Poisson–Boltzmann Equation"
5713:
5300:
530:{\displaystyle c_{i}=c_{i}^{0}\cdot e^{\frac {-W_{i}}{k_{\mathrm {B} }T}},}
52:
The Poisson–Boltzmann equation describes a model proposed independently by
5663:
5219:
5830:
5202:
4384:(MIS). This can be used to describe both time and position dependence of
4361:
3430:
5894:
2084:{\displaystyle \lambda _{D}={\frac {\mathrm {0.304nm} }{\sqrt {c_{0}}}}}
5883:
3405:
The linearized Poisson–Boltzmann equation can be used to calculate the
5897:
Adaptive Fast Multipole Poisson–Boltzmann Solver, free and open-source
5695:
5531:
3509:{\displaystyle \Delta G^{\text{el}}=\int ^{\tau }qU(\tau ')\,d\tau '}
659:
5292:
437:
The freedom of movement of ions in solution can be accounted for by
5871:
3927:{\displaystyle \Delta G^{\text{ef}}=\int _{V}{\frac {p^{f}U}{2}}dV}
3762:
Note that the free energy is independent of the charging pathway .
1203:; however, with certain geometries, it can be solved analytically.
934:{\displaystyle c^{+}=c_{0}\cdot e^{\frac {-e\psi (x,y,z)}{k_{B}T}}}
3374:
1794:{\displaystyle e\left\vert \psi \right\vert \ll k_{\mathrm {B} }T}
835:{\displaystyle c^{-}=c_{0}\cdot e^{\frac {e\psi (x,y,z)}{k_{B}T}}}
129:
103:
Thermal motion of ions accounted for; ions behave as point charges
2032:
water with a monovalent salt. The Debye length equation is then:
5434:
Department of Chemical Engineering, Carnegie Mellon University.
3410:
5886:
Matched Interface & Boundary based Poisson–Boltzmann solver
5677:
Gruziel, Magdalena; Grochowski, Pawel; Trylska, Joanna (2008).
3024:, substitute this for y in the previous equation and solve for
1641:{\displaystyle R_{eD}={\sqrt {\frac {kT}{4\pi e^{2}n_{e0}}}}}
1153:{\displaystyle \rho _{e}=e{(c^{+}-c^{-})}=c_{0}e\cdot \left.}
4866:
4771:
4725:
4411:
4286:
4140:
4038:
3942:
3869:
3771:
3553:
3522:
3447:
3302:
3052:
3030:
2980:
2754:
2520:
2483:
2301:
2245:
2128:
2099:
2040:
2008:
1961:
1874:
1845:
1807:
1758:
1700:
1654:
1576:
1533:
1415:
1220:
958:
848:
752:
713:
677:
644:
609:
580:
546:
451:
445:
is used to calculate the local ion density such that
402:
369:
340:
151:
3409:
and free energy of highly charged molecules such as
4032:Entropic free energy of mixing of mobile species =
5107:
4850:
4754:
4708:
4350:
4269:
4124:
4022:
3926:
3851:
3752:
3531:
3508:
3353:{\displaystyle y_{0}={\frac {e\psi _{0}}{k_{B}T}}}
3352:
3287:
3036:
3016:
2964:
2736:
2504:
2467:
2281:
2219:
2112:
2083:
2024:
1992:
1946:
1858:
1827:
1793:
1737:
1670:
1640:
1562:
1517:
1399:
1152:
933:
834:
738:
699:
650:
624:
593:
564:
529:
415:
382:
353:
323:
1738:{\displaystyle \psi =\psi _{0}e^{-\mathrm {K} x}}
1175:as Gouy-Chapman theory; in solution chemistry as
5590:Cell Physiology Sourcebook: A Molecular Approach
3017:{\displaystyle y\equiv {\frac {e\psi }{k_{B}T}}}
2282:{\displaystyle y\equiv {\frac {e\psi }{k_{B}T}}}
134:A simple planar case for the Gouy–Chapman model
5357:(2nd ed.). Weinheim, Germany: Wiley-VCH.
4392:and self-consistent average potential for the
1993:{\displaystyle \lambda _{D}=\mathrm {K} ^{-1}}
1185:Derjaguin–Landau–Verwey–Overbeek (DLVO) theory
138:The Poisson–Boltzmann equation describes the
8:
4134:Entropic free energy of mixing of solvent =
1195:Because the Poisson–Boltzmann equation is a
361:is the local electric charge density in C/m,
5880:– A Poisson–Boltzmann electrostatics solver
5376:
5374:
5348:
5346:
5344:
5342:
5340:
5338:
1199:of the second order, it is commonly solved
5336:
5334:
5332:
5330:
5328:
5326:
5324:
5322:
5320:
5318:
1563:{\displaystyle \psi ={\frac {e\Phi }{kT}}}
5829:
5752:
5703:
5653:
5201:
5088:
5078:
5063:
5053:
5039:
5032:
5005:
4995:
4985:
4974:
4955:
4940:
4926:
4925:
4909:
4889:
4880:
4871:
4865:
4834:
4826:
4810:
4795:
4785:
4776:
4770:
4746:
4733:
4724:
4678:
4668:
4614:
4592:
4588:
4575:
4569:
4564:
4543:
4529:
4505:
4490:
4480:
4468:
4458:
4448:
4439:
4426:
4416:
4410:
4342:
4326:
4310:
4294:
4285:
4229:
4219:
4192:
4187:
4177:
4167:
4148:
4139:
4108:
4103:
4093:
4087:
4075:
4065:
4046:
4037:
3996:
3986:
3976:
3969:
3963:
3950:
3941:
3903:
3896:
3890:
3877:
3868:
3843:
3827:
3811:
3795:
3779:
3770:
3722:
3710:
3698:
3686:
3647:
3637:
3610:
3605:
3595:
3574:
3561:
3552:
3521:
3494:
3468:
3455:
3446:
3338:
3326:
3316:
3307:
3301:
3272:
3268:
3242:
3236:
3231:
3205:
3199:
3194:
3178:
3174:
3148:
3142:
3137:
3111:
3105:
3100:
3093:
3069:
3059:
3051:
3029:
3002:
2987:
2979:
2949:
2945:
2919:
2913:
2908:
2882:
2876:
2871:
2855:
2851:
2825:
2819:
2814:
2788:
2782:
2777:
2770:
2753:
2721:
2717:
2691:
2685:
2680:
2654:
2648:
2643:
2627:
2623:
2597:
2591:
2586:
2560:
2554:
2549:
2542:
2529:
2525:
2519:
2492:
2488:
2482:
2443:
2437:
2432:
2406:
2402:
2374:
2368:
2363:
2337:
2333:
2323:
2310:
2306:
2300:
2267:
2252:
2244:
2209:
2202:
2197:
2190:
2170:
2169:
2159:
2145:
2139:
2137:
2129:
2127:
2104:
2098:
2072:
2056:
2054:
2045:
2039:
2013:
2007:
1981:
1976:
1966:
1960:
1930:
1929:
1919:
1904:
1894:
1883:
1875:
1873:
1850:
1844:
1828:{\displaystyle \psi \leq \mathrm {25mV} }
1814:
1806:
1781:
1780:
1757:
1725:
1721:
1711:
1699:
1659:
1653:
1625:
1615:
1593:
1581:
1575:
1540:
1532:
1506:
1490:
1463:
1453:
1441:
1423:
1416:
1414:
1378:
1377:
1350:
1329:
1328:
1304:
1283:
1265:
1258:
1246:
1228:
1221:
1219:
1129:
1093:
1073:
1034:
1013:
996:
983:
975:
963:
957:
918:
879:
866:
853:
847:
819:
783:
770:
757:
751:
718:
712:
682:
676:
643:
615:
614:
608:
585:
579:
556:
551:
545:
510:
509:
497:
487:
474:
469:
456:
450:
407:
401:
374:
368:
345:
339:
309:
299:
288:
282:
267:
249:
242:
230:
212:
205:
193:
175:
168:
156:
150:
5555:Journal of Colloid and Interface Science
4755:{\displaystyle \rho \rho _{1}+\rho _{2}}
74:
5429:
5427:
5180:
60:in 1910 and 1913, respectively. In the
5679:"The Poisson-Boltzmann model for tRNA"
5353:Butt, H.; Graf, L.; Kappl, M. (2006).
3044:. The following equation is rendered.
5768:
5766:
5764:
5613:
5611:
5609:
5401:
5399:
1686:Low-potential vs high-potential cases
572:is the ion concentration at the bulk,
7:
5481:
5479:
5274:
5272:
123:The Gouy–Chapman model explains the
5355:Physics and Chemistry of Interfaces
3425:. This takes into account both the
1752:Strictly, low potential means that
945:is the charge of an electron, 1.602
4686:
4671:
4624:
4498:
4483:
4335:
4319:
4303:
4287:
4193:
4141:
4109:
4039:
3943:
3870:
3836:
3820:
3804:
3788:
3772:
3611:
3554:
3539:is the final charge on the sphere
3448:
3273:
3179:
2950:
2856:
2722:
2628:
2311:
2171:
2130:
2063:
2060:
1977:
1931:
1876:
1821:
1818:
1782:
1726:
1546:
1527:It uses a dimensionless potential
1379:
1330:
616:
511:
423:is the permittivity of free space,
260:
246:
223:
209:
186:
172:
153:
25:
5872:Adaptive Poisson–Boltzmann Solver
5592:(3rd ed.). San Diego: Acad.
3711:
625:{\displaystyle k_{\mathrm {B} }}
416:{\displaystyle \varepsilon _{0}}
383:{\displaystyle \varepsilon _{r}}
5754:10.1590/S0103-97332000000200023
5486:Lu, B. Z.; et al. (2008).
5255:. Academic Press. p. 318.
5190:The European Physical Journal E
3936:Electrostatic mobile charges =
5921:Eponymous equations of physics
4918:
4899:
4719:the system can be found using
4662:
4659:
4642:
4630:
3863:Electrostatic fixed charges =
3719:
3707:
3491:
3480:
3258:
3224:
3164:
3130:
2935:
2901:
2841:
2807:
2707:
2673:
2613:
2579:
2459:
2425:
2422:
2395:
2390:
2356:
2353:
2326:
1368:
1362:
1319:
1313:
1120:
1102:
1064:
1046:
1002:
976:
909:
891:
810:
792:
1:
5646:10.1016/S0006-3495(99)77173-0
5575:10.1016/S0021-9797(02)00142-X
5381:New Mexico State University.
1748:Low-potential case conditions
739:{\displaystyle W^{-}=-e\psi }
5733:Brazilian Journal of Physics
2025:{\displaystyle 25^{\circ }C}
1859:{\displaystyle \lambda _{D}}
700:{\displaystyle W^{+}=e\psi }
390:is the dielectric constant (
5249:Attard, Phil (2002-08-07).
3402:electrostatic free energy.
5947:
5795:10.1088/1674-1056/18/7/059
5408:"Chemistry 465 Lecture 10"
3397:Physiological applications
32:Poisson–Boltzmann equation
18:Poisson-Boltzmann equation
5848:10.1209/epl/i2000-00495-1
5512:Technical Physics Letters
5406:Simon Fraser University.
3437:Electrostatic free energy
2477:This equation solved for
565:{\displaystyle c_{i}^{0}}
354:{\displaystyle \rho _{e}}
140:electrochemical potential
81:Important characteristics
48:Background and derivation
2291:trigonometric identities
5890:CHARMM-GUI: PBEQ Solver
5588:Sperelakis, N. (2012).
5460:"Electric Double Layer"
5383:"Electric Double Layer"
3407:electrostatic potential
2505:{\displaystyle e^{y/2}}
5109:
4852:
4756:
4710:
4352:
4271:
4126:
4024:
3928:
3853:
3754:
3533:
3532:{\displaystyle \tau q}
3510:
3380:
3354:
3289:
3038:
3018:
2966:
2738:
2506:
2469:
2283:
2221:
2114:
2085:
2026:
1994:
1948:
1860:
1829:
1795:
1739:
1672:
1671:{\displaystyle n_{e0}}
1642:
1564:
1519:
1401:
1154:
935:
836:
740:
701:
671:can be represented by
658:is the temperature in
652:
626:
595:
566:
531:
417:
384:
355:
325:
135:
5493:Commun. Comput. Phys.
5220:10.1007/s101890050023
5110:
4853:
4757:
4711:
4353:
4272:
4127:
4025:
3929:
3854:
3755:
3534:
3511:
3378:
3355:
3290:
3039:
3037:{\displaystyle \psi }
3019:
2967:
2739:
2507:
2470:
2284:
2222:
2115:
2113:{\displaystyle c_{0}}
2086:
2027:
1995:
1949:
1861:
1830:
1796:
1740:
1673:
1643:
1565:
1520:
1402:
1155:
936:
837:
741:
702:
653:
627:
596:
594:{\displaystyle W_{i}}
567:
532:
418:
392:relative permittivity
385:
356:
326:
133:
58:David Leonard Chapman
4864:
4769:
4723:
4409:
4284:
4138:
4036:
3940:
3867:
3769:
3551:
3520:
3445:
3419:phospholipid bilayer
3384:General applications
3300:
3050:
3028:
2978:
2752:
2518:
2481:
2299:
2243:
2126:
2097:
2038:
2006:
1959:
1872:
1843:
1805:
1756:
1698:
1652:
1574:
1531:
1413:
1218:
1197:partial differential
1191:Solving analytically
956:
846:
750:
711:
675:
642:
607:
578:
544:
449:
439:Boltzmann statistics
400:
367:
338:
149:
5931:Colloidal chemistry
5840:2000EL.....52..705M
5818:Europhysics Letters
5787:2009ChPhB..18.2975Z
5745:2000BrJPh..30..403C
5638:1999BpJ....76....1F
5625:Biophysical Journal
5567:2003JCIS..258...45T
5524:2005TePhL..31..204D
5212:2000EPJE....1..203N
4574:
4386:dissipative systems
4197:
4113:
3615:
3389:further insight on
2230:High-potential case
1177:Debye–Huckel theory
561:
479:
5926:Molecular dynamics
5905:Philip T. Gressman
5105:
4848:
4752:
4706:
4560:
4402:electronic density
4390:Boltzmann equation
4348:
4267:
4183:
4182:
4122:
4099:
4020:
3981:
3924:
3849:
3750:
3601:
3600:
3529:
3506:
3381:
3350:
3285:
3034:
3014:
2962:
2734:
2502:
2465:
2279:
2217:
2110:
2081:
2022:
1990:
1944:
1856:
1825:
1791:
1735:
1668:
1638:
1560:
1515:
1397:
1150:
931:
832:
736:
697:
648:
634:Boltzmann constant
622:
591:
562:
547:
527:
465:
443:Boltzmann equation
431:electric potential
413:
380:
351:
321:
136:
62:Gouy-Chapman model
54:Louis Georges Gouy
5775:Chinese Physics B
5696:10.1002/jcc.20953
5690:(12): 1970–1981.
5599:978-0-12-387738-3
5532:10.1134/1.1894433
5364:978-3-527-40629-6
5262:978-0-12-066321-7
5096:
5095:
5015:
4979:
4946:
4894:
4820:
4693:
4620:
4583:
4549:
4512:
4478:
4376:Materials science
4370:colloid chemistry
4345:
4329:
4313:
4297:
4250:
4173:
4151:
4114:
4049:
4012:
3972:
3953:
3916:
3880:
3846:
3830:
3814:
3798:
3782:
3737:
3668:
3591:
3564:
3458:
3348:
3283:
3082:
3012:
2960:
2732:
2463:
2277:
2215:
2181:
2079:
2078:
1942:
1941:
1636:
1635:
1558:
1481:
1461:
1448:
1389:
1340:
1290:
1253:
1181:colloid chemistry
1139:
1083:
928:
829:
651:{\displaystyle T}
521:
394:) of the solvent,
316:
274:
237:
200:
121:
120:
70:Hermann Helmholtz
16:(Redirected from
5938:
5860:
5859:
5833:
5831:cond-mat/0009376
5813:
5807:
5806:
5781:(2): 2975–2980.
5770:
5759:
5758:
5756:
5724:
5718:
5717:
5707:
5684:J. Comput. Chem.
5674:
5668:
5667:
5657:
5615:
5604:
5603:
5585:
5579:
5578:
5550:
5544:
5543:
5507:
5501:
5500:
5483:
5474:
5473:
5471:
5470:
5456:
5450:
5449:
5447:
5445:
5440:
5431:
5422:
5421:
5419:
5417:
5412:
5403:
5394:
5393:
5391:
5389:
5378:
5369:
5368:
5350:
5313:
5312:
5281:J. Mol. Recognit
5276:
5267:
5266:
5246:
5240:
5239:
5205:
5203:cond-mat/9902085
5185:
5145:
5144:
5140:
5120:
5114:
5112:
5111:
5106:
5104:
5100:
5099:
5098:
5097:
5094:
5093:
5092:
5083:
5082:
5069:
5068:
5067:
5058:
5057:
5041:
5040:
5016:
5014:
5013:
5012:
5000:
4999:
4990:
4989:
4980:
4975:
4969:
4968:
4967:
4963:
4962:
4947:
4945:
4944:
4935:
4927:
4917:
4916:
4895:
4890:
4881:
4876:
4875:
4857:
4855:
4854:
4849:
4847:
4846:
4842:
4841:
4821:
4819:
4818:
4817:
4801:
4800:
4799:
4786:
4781:
4780:
4761:
4759:
4758:
4753:
4751:
4750:
4738:
4737:
4715:
4713:
4712:
4707:
4705:
4694:
4692:
4684:
4683:
4682:
4669:
4658:
4623:
4622:
4621:
4619:
4618:
4609:
4608:
4593:
4584:
4576:
4573:
4568:
4556:
4552:
4551:
4550:
4548:
4547:
4538:
4530:
4513:
4511:
4510:
4509:
4496:
4495:
4494:
4481:
4479:
4474:
4473:
4472:
4463:
4462:
4449:
4444:
4443:
4431:
4430:
4421:
4420:
4398:electric current
4394:Poisson equation
4357:
4355:
4354:
4349:
4347:
4346:
4343:
4331:
4330:
4327:
4315:
4314:
4311:
4299:
4298:
4295:
4276:
4274:
4273:
4268:
4260:
4256:
4255:
4251:
4249:
4241:
4234:
4233:
4220:
4196:
4191:
4181:
4172:
4171:
4153:
4152:
4149:
4131:
4129:
4128:
4123:
4115:
4112:
4107:
4098:
4097:
4088:
4080:
4079:
4070:
4069:
4051:
4050:
4047:
4029:
4027:
4026:
4021:
4013:
4008:
4001:
4000:
3991:
3990:
3980:
3970:
3968:
3967:
3955:
3954:
3951:
3933:
3931:
3930:
3925:
3917:
3912:
3908:
3907:
3897:
3895:
3894:
3882:
3881:
3878:
3858:
3856:
3855:
3850:
3848:
3847:
3844:
3832:
3831:
3828:
3816:
3815:
3812:
3800:
3799:
3796:
3784:
3783:
3780:
3759:
3757:
3756:
3751:
3743:
3739:
3738:
3736:
3728:
3727:
3726:
3714:
3699:
3691:
3690:
3678:
3674:
3673:
3669:
3667:
3659:
3652:
3651:
3638:
3614:
3609:
3599:
3579:
3578:
3566:
3565:
3562:
3538:
3536:
3535:
3530:
3515:
3513:
3512:
3507:
3505:
3490:
3473:
3472:
3460:
3459:
3456:
3359:
3357:
3356:
3351:
3349:
3347:
3343:
3342:
3332:
3331:
3330:
3317:
3312:
3311:
3294:
3292:
3291:
3286:
3284:
3282:
3281:
3280:
3276:
3251:
3250:
3246:
3241:
3240:
3214:
3213:
3209:
3204:
3203:
3188:
3187:
3186:
3182:
3157:
3156:
3152:
3147:
3146:
3120:
3119:
3115:
3110:
3109:
3094:
3083:
3078:
3074:
3073:
3060:
3043:
3041:
3040:
3035:
3023:
3021:
3020:
3015:
3013:
3011:
3007:
3006:
2996:
2988:
2971:
2969:
2968:
2963:
2961:
2959:
2958:
2957:
2953:
2928:
2927:
2923:
2918:
2917:
2891:
2890:
2886:
2881:
2880:
2865:
2864:
2863:
2859:
2834:
2833:
2829:
2824:
2823:
2797:
2796:
2792:
2787:
2786:
2771:
2743:
2741:
2740:
2735:
2733:
2731:
2730:
2729:
2725:
2700:
2699:
2695:
2690:
2689:
2663:
2662:
2658:
2653:
2652:
2637:
2636:
2635:
2631:
2606:
2605:
2601:
2596:
2595:
2569:
2568:
2564:
2559:
2558:
2543:
2538:
2537:
2533:
2512:is shown below.
2511:
2509:
2508:
2503:
2501:
2500:
2496:
2474:
2472:
2471:
2466:
2464:
2462:
2452:
2451:
2447:
2442:
2441:
2415:
2414:
2410:
2393:
2383:
2382:
2378:
2373:
2372:
2346:
2345:
2341:
2324:
2319:
2318:
2314:
2288:
2286:
2285:
2280:
2278:
2276:
2272:
2271:
2261:
2253:
2226:
2224:
2223:
2218:
2216:
2214:
2213:
2208:
2207:
2206:
2195:
2194:
2182:
2180:
2176:
2175:
2174:
2164:
2163:
2150:
2149:
2140:
2138:
2133:
2119:
2117:
2116:
2111:
2109:
2108:
2090:
2088:
2087:
2082:
2080:
2077:
2076:
2067:
2066:
2055:
2050:
2049:
2031:
2029:
2028:
2023:
2018:
2017:
1999:
1997:
1996:
1991:
1989:
1988:
1980:
1971:
1970:
1953:
1951:
1950:
1945:
1943:
1940:
1936:
1935:
1934:
1924:
1923:
1910:
1909:
1908:
1899:
1898:
1885:
1884:
1879:
1865:
1863:
1862:
1857:
1855:
1854:
1834:
1832:
1831:
1826:
1824:
1800:
1798:
1797:
1792:
1787:
1786:
1785:
1772:
1744:
1742:
1741:
1736:
1734:
1733:
1729:
1716:
1715:
1677:
1675:
1674:
1669:
1667:
1666:
1647:
1645:
1644:
1639:
1637:
1634:
1633:
1632:
1620:
1619:
1603:
1595:
1594:
1589:
1588:
1569:
1567:
1566:
1561:
1559:
1557:
1549:
1541:
1524:
1522:
1521:
1516:
1514:
1513:
1495:
1494:
1482:
1480:
1472:
1464:
1462:
1454:
1449:
1447:
1446:
1445:
1432:
1428:
1427:
1417:
1406:
1404:
1403:
1398:
1396:
1392:
1391:
1390:
1388:
1384:
1383:
1382:
1371:
1351:
1342:
1341:
1339:
1335:
1334:
1333:
1322:
1305:
1291:
1289:
1288:
1287:
1274:
1270:
1269:
1259:
1254:
1252:
1251:
1250:
1237:
1233:
1232:
1222:
1173:electrochemistry
1166:Related theories
1159:
1157:
1156:
1151:
1146:
1142:
1141:
1140:
1138:
1134:
1133:
1123:
1094:
1085:
1084:
1082:
1078:
1077:
1067:
1035:
1018:
1017:
1005:
1001:
1000:
988:
987:
968:
967:
948:
940:
938:
937:
932:
930:
929:
927:
923:
922:
912:
880:
871:
870:
858:
857:
841:
839:
838:
833:
831:
830:
828:
824:
823:
813:
784:
775:
774:
762:
761:
745:
743:
742:
737:
723:
722:
706:
704:
703:
698:
687:
686:
670:
657:
655:
654:
649:
631:
629:
628:
623:
621:
620:
619:
600:
598:
597:
592:
590:
589:
571:
569:
568:
563:
560:
555:
536:
534:
533:
528:
523:
522:
520:
516:
515:
514:
503:
502:
501:
488:
478:
473:
461:
460:
428:
422:
420:
419:
414:
412:
411:
389:
387:
386:
381:
379:
378:
360:
358:
357:
352:
350:
349:
330:
328:
327:
322:
317:
315:
314:
313:
304:
303:
293:
292:
283:
275:
273:
272:
271:
258:
254:
253:
243:
238:
236:
235:
234:
221:
217:
216:
206:
201:
199:
198:
197:
184:
180:
179:
169:
161:
160:
144:Poisson equation
75:
21:
5946:
5945:
5941:
5940:
5939:
5937:
5936:
5935:
5911:
5910:
5868:
5863:
5815:
5814:
5810:
5772:
5771:
5762:
5726:
5725:
5721:
5676:
5675:
5671:
5617:
5616:
5607:
5600:
5587:
5586:
5582:
5552:
5551:
5547:
5509:
5508:
5504:
5499:(5): 973–1009 .
5485:
5484:
5477:
5468:
5466:
5458:
5457:
5453:
5443:
5441:
5438:
5433:
5432:
5425:
5415:
5413:
5410:
5405:
5404:
5397:
5387:
5385:
5380:
5379:
5372:
5365:
5352:
5351:
5316:
5293:10.1002/jmr.577
5278:
5277:
5270:
5263:
5248:
5247:
5243:
5187:
5186:
5182:
5178:
5166:
5146:
5142:
5138:
5136:
5135:
5118:
5084:
5074:
5070:
5059:
5049:
5042:
5028:
5021:
5017:
5001:
4991:
4981:
4970:
4951:
4936:
4928:
4921:
4905:
4882:
4867:
4862:
4861:
4830:
4822:
4806:
4802:
4791:
4787:
4772:
4767:
4766:
4742:
4729:
4721:
4720:
4698:
4685:
4674:
4670:
4651:
4610:
4601:
4594:
4589:
4539:
4531:
4525:
4518:
4514:
4501:
4497:
4486:
4482:
4464:
4454:
4450:
4435:
4422:
4412:
4407:
4406:
4378:
4338:
4322:
4306:
4290:
4282:
4281:
4242:
4225:
4221:
4215:
4202:
4198:
4163:
4144:
4136:
4135:
4089:
4071:
4061:
4042:
4034:
4033:
3992:
3982:
3971:
3959:
3946:
3938:
3937:
3899:
3898:
3886:
3873:
3865:
3864:
3839:
3823:
3807:
3791:
3775:
3767:
3766:
3729:
3718:
3700:
3682:
3660:
3643:
3639:
3633:
3620:
3616:
3584:
3580:
3570:
3557:
3549:
3548:
3518:
3517:
3498:
3483:
3464:
3451:
3443:
3442:
3439:
3399:
3386:
3365:
3334:
3333:
3322:
3318:
3303:
3298:
3297:
3264:
3232:
3227:
3195:
3190:
3189:
3170:
3138:
3133:
3101:
3096:
3095:
3065:
3061:
3048:
3047:
3026:
3025:
2998:
2997:
2989:
2976:
2975:
2941:
2909:
2904:
2872:
2867:
2866:
2847:
2815:
2810:
2778:
2773:
2772:
2750:
2749:
2713:
2681:
2676:
2644:
2639:
2638:
2619:
2587:
2582:
2550:
2545:
2544:
2521:
2516:
2515:
2484:
2479:
2478:
2433:
2428:
2398:
2394:
2364:
2359:
2329:
2325:
2302:
2297:
2296:
2263:
2262:
2254:
2241:
2240:
2232:
2198:
2196:
2186:
2165:
2155:
2151:
2141:
2124:
2123:
2100:
2095:
2094:
2068:
2041:
2036:
2035:
2009:
2004:
2003:
1975:
1962:
1957:
1956:
1925:
1915:
1911:
1900:
1890:
1886:
1870:
1869:
1846:
1841:
1840:
1803:
1802:
1776:
1762:
1754:
1753:
1750:
1717:
1707:
1696:
1695:
1688:
1655:
1650:
1649:
1621:
1611:
1604:
1596:
1577:
1572:
1571:
1550:
1542:
1529:
1528:
1502:
1486:
1473:
1465:
1437:
1433:
1419:
1418:
1411:
1410:
1373:
1372:
1352:
1346:
1324:
1323:
1306:
1300:
1299:
1295:
1279:
1275:
1261:
1260:
1242:
1238:
1224:
1223:
1216:
1215:
1209:
1193:
1168:
1125:
1124:
1095:
1089:
1069:
1068:
1036:
1030:
1029:
1025:
1009:
992:
979:
959:
954:
953:
946:
914:
913:
881:
875:
862:
849:
844:
843:
815:
814:
785:
779:
766:
753:
748:
747:
714:
709:
708:
678:
673:
672:
668:
640:
639:
610:
605:
604:
581:
576:
575:
542:
541:
505:
504:
493:
489:
483:
452:
447:
446:
426:
403:
398:
397:
370:
365:
364:
341:
336:
335:
305:
295:
294:
284:
263:
259:
245:
244:
226:
222:
208:
207:
189:
185:
171:
170:
152:
147:
146:
50:
45:
28:
23:
22:
15:
12:
11:
5:
5944:
5942:
5934:
5933:
5928:
5923:
5913:
5912:
5909:
5908:
5898:
5892:
5887:
5881:
5875:
5867:
5866:External links
5864:
5862:
5861:
5824:(6): 705–711.
5808:
5760:
5739:(2): 403–409.
5719:
5669:
5605:
5598:
5580:
5545:
5518:(3): 204–207.
5502:
5475:
5451:
5423:
5395:
5370:
5363:
5314:
5287:(6): 379–385.
5268:
5261:
5241:
5196:(2): 203–214.
5179:
5177:
5174:
5173:
5172:
5165:
5162:
5134:
5131:
5121:is called the
5116:
5115:
5103:
5091:
5087:
5081:
5077:
5073:
5066:
5062:
5056:
5052:
5048:
5045:
5038:
5035:
5031:
5027:
5024:
5020:
5011:
5008:
5004:
4998:
4994:
4988:
4984:
4978:
4973:
4966:
4961:
4958:
4954:
4950:
4943:
4939:
4934:
4931:
4924:
4920:
4915:
4912:
4908:
4904:
4901:
4898:
4893:
4888:
4885:
4879:
4874:
4870:
4859:
4845:
4840:
4837:
4833:
4829:
4825:
4816:
4813:
4809:
4805:
4798:
4794:
4790:
4784:
4779:
4775:
4749:
4745:
4741:
4736:
4732:
4728:
4704:
4701:
4697:
4691:
4688:
4681:
4677:
4673:
4667:
4664:
4661:
4657:
4654:
4650:
4647:
4644:
4641:
4638:
4635:
4632:
4629:
4626:
4617:
4613:
4607:
4604:
4600:
4597:
4591:
4587:
4582:
4579:
4572:
4567:
4563:
4559:
4555:
4546:
4542:
4537:
4534:
4528:
4524:
4521:
4517:
4508:
4504:
4500:
4493:
4489:
4485:
4477:
4471:
4467:
4461:
4457:
4453:
4447:
4442:
4438:
4434:
4429:
4425:
4419:
4415:
4377:
4374:
4341:
4337:
4334:
4325:
4321:
4318:
4309:
4305:
4302:
4293:
4289:
4278:
4277:
4266:
4263:
4259:
4254:
4248:
4245:
4240:
4237:
4232:
4228:
4224:
4218:
4214:
4211:
4208:
4205:
4201:
4195:
4190:
4186:
4180:
4176:
4170:
4166:
4162:
4159:
4156:
4147:
4143:
4132:
4121:
4118:
4111:
4106:
4102:
4096:
4092:
4086:
4083:
4078:
4074:
4068:
4064:
4060:
4057:
4054:
4045:
4041:
4030:
4019:
4016:
4011:
4007:
4004:
3999:
3995:
3989:
3985:
3979:
3975:
3966:
3962:
3958:
3949:
3945:
3934:
3923:
3920:
3915:
3911:
3906:
3902:
3893:
3889:
3885:
3876:
3872:
3842:
3838:
3835:
3826:
3822:
3819:
3810:
3806:
3803:
3794:
3790:
3787:
3778:
3774:
3749:
3746:
3742:
3735:
3732:
3725:
3721:
3717:
3713:
3709:
3706:
3703:
3697:
3694:
3689:
3685:
3681:
3677:
3672:
3666:
3663:
3658:
3655:
3650:
3646:
3642:
3636:
3632:
3629:
3626:
3623:
3619:
3613:
3608:
3604:
3598:
3594:
3590:
3587:
3583:
3577:
3573:
3569:
3560:
3556:
3545:Euler-Lagrange
3528:
3525:
3504:
3501:
3497:
3493:
3489:
3486:
3482:
3479:
3476:
3471:
3467:
3463:
3454:
3450:
3438:
3435:
3398:
3395:
3391:electrostatics
3385:
3382:
3364:
3361:
3346:
3341:
3337:
3329:
3325:
3321:
3315:
3310:
3306:
3279:
3275:
3271:
3267:
3263:
3260:
3257:
3254:
3249:
3245:
3239:
3235:
3230:
3226:
3223:
3220:
3217:
3212:
3208:
3202:
3198:
3193:
3185:
3181:
3177:
3173:
3169:
3166:
3163:
3160:
3155:
3151:
3145:
3141:
3136:
3132:
3129:
3126:
3123:
3118:
3114:
3108:
3104:
3099:
3092:
3089:
3086:
3081:
3077:
3072:
3068:
3064:
3058:
3055:
3033:
3010:
3005:
3001:
2995:
2992:
2986:
2983:
2956:
2952:
2948:
2944:
2940:
2937:
2934:
2931:
2926:
2922:
2916:
2912:
2907:
2903:
2900:
2897:
2894:
2889:
2885:
2879:
2875:
2870:
2862:
2858:
2854:
2850:
2846:
2843:
2840:
2837:
2832:
2828:
2822:
2818:
2813:
2809:
2806:
2803:
2800:
2795:
2791:
2785:
2781:
2776:
2769:
2766:
2763:
2760:
2757:
2728:
2724:
2720:
2716:
2712:
2709:
2706:
2703:
2698:
2694:
2688:
2684:
2679:
2675:
2672:
2669:
2666:
2661:
2657:
2651:
2647:
2642:
2634:
2630:
2626:
2622:
2618:
2615:
2612:
2609:
2604:
2600:
2594:
2590:
2585:
2581:
2578:
2575:
2572:
2567:
2563:
2557:
2553:
2548:
2541:
2536:
2532:
2528:
2524:
2499:
2495:
2491:
2487:
2461:
2458:
2455:
2450:
2446:
2440:
2436:
2431:
2427:
2424:
2421:
2418:
2413:
2409:
2405:
2401:
2397:
2392:
2389:
2386:
2381:
2377:
2371:
2367:
2362:
2358:
2355:
2352:
2349:
2344:
2340:
2336:
2332:
2328:
2322:
2317:
2313:
2309:
2305:
2275:
2270:
2266:
2260:
2257:
2251:
2248:
2231:
2228:
2212:
2205:
2201:
2193:
2189:
2185:
2179:
2173:
2168:
2162:
2158:
2154:
2148:
2144:
2136:
2132:
2107:
2103:
2075:
2071:
2065:
2062:
2059:
2053:
2048:
2044:
2021:
2016:
2012:
1987:
1984:
1979:
1974:
1969:
1965:
1939:
1933:
1928:
1922:
1918:
1914:
1907:
1903:
1897:
1893:
1889:
1882:
1878:
1853:
1849:
1823:
1820:
1817:
1813:
1810:
1790:
1784:
1779:
1775:
1771:
1768:
1765:
1761:
1749:
1746:
1732:
1728:
1724:
1720:
1714:
1710:
1706:
1703:
1687:
1684:
1680:number density
1665:
1662:
1658:
1631:
1628:
1624:
1618:
1614:
1610:
1607:
1602:
1599:
1592:
1587:
1584:
1580:
1556:
1553:
1548:
1545:
1539:
1536:
1512:
1509:
1505:
1501:
1498:
1493:
1489:
1485:
1479:
1476:
1471:
1468:
1460:
1457:
1452:
1444:
1440:
1436:
1431:
1426:
1422:
1395:
1387:
1381:
1376:
1370:
1367:
1364:
1361:
1358:
1355:
1349:
1345:
1338:
1332:
1327:
1321:
1318:
1315:
1312:
1309:
1303:
1298:
1294:
1286:
1282:
1278:
1273:
1268:
1264:
1257:
1249:
1245:
1241:
1236:
1231:
1227:
1208:
1205:
1192:
1189:
1167:
1164:
1149:
1145:
1137:
1132:
1128:
1122:
1119:
1116:
1113:
1110:
1107:
1104:
1101:
1098:
1092:
1088:
1081:
1076:
1072:
1066:
1063:
1060:
1057:
1054:
1051:
1048:
1045:
1042:
1039:
1033:
1028:
1024:
1021:
1016:
1012:
1008:
1004:
999:
995:
991:
986:
982:
978:
974:
971:
966:
962:
926:
921:
917:
911:
908:
905:
902:
899:
896:
893:
890:
887:
884:
878:
874:
869:
865:
861:
856:
852:
827:
822:
818:
812:
809:
806:
803:
800:
797:
794:
791:
788:
782:
778:
773:
769:
765:
760:
756:
735:
732:
729:
726:
721:
717:
696:
693:
690:
685:
681:
664:
663:
647:
637:
618:
613:
602:
588:
584:
573:
559:
554:
550:
526:
519:
513:
508:
500:
496:
492:
486:
482:
477:
472:
468:
464:
459:
455:
435:
434:
424:
410:
406:
395:
377:
373:
362:
348:
344:
320:
312:
308:
302:
298:
291:
287:
281:
278:
270:
266:
262:
257:
252:
248:
241:
233:
229:
225:
220:
215:
211:
204:
196:
192:
188:
183:
178:
174:
167:
164:
159:
155:
119:
118:
115:
112:
108:
107:
104:
101:
97:
96:
93:
90:
86:
85:
82:
79:
49:
46:
44:
41:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
5943:
5932:
5929:
5927:
5924:
5922:
5919:
5918:
5916:
5906:
5902:
5899:
5896:
5893:
5891:
5888:
5885:
5882:
5879:
5876:
5873:
5870:
5869:
5865:
5857:
5853:
5849:
5845:
5841:
5837:
5832:
5827:
5823:
5819:
5812:
5809:
5804:
5800:
5796:
5792:
5788:
5784:
5780:
5776:
5769:
5767:
5765:
5761:
5755:
5750:
5746:
5742:
5738:
5734:
5730:
5723:
5720:
5715:
5711:
5706:
5701:
5697:
5693:
5689:
5686:
5685:
5680:
5673:
5670:
5665:
5661:
5656:
5651:
5647:
5643:
5639:
5635:
5631:
5627:
5626:
5621:
5614:
5612:
5610:
5606:
5601:
5595:
5591:
5584:
5581:
5576:
5572:
5568:
5564:
5560:
5556:
5549:
5546:
5541:
5537:
5533:
5529:
5525:
5521:
5517:
5513:
5506:
5503:
5498:
5495:
5494:
5489:
5482:
5480:
5476:
5465:
5461:
5455:
5452:
5437:
5430:
5428:
5424:
5409:
5402:
5400:
5396:
5384:
5377:
5375:
5371:
5366:
5360:
5356:
5349:
5347:
5345:
5343:
5341:
5339:
5337:
5335:
5333:
5331:
5329:
5327:
5325:
5323:
5321:
5319:
5315:
5310:
5306:
5302:
5298:
5294:
5290:
5286:
5282:
5275:
5273:
5269:
5264:
5258:
5254:
5253:
5245:
5242:
5237:
5233:
5229:
5225:
5221:
5217:
5213:
5209:
5204:
5199:
5195:
5191:
5184:
5181:
5175:
5171:
5168:
5167:
5163:
5161:
5158:
5154:
5152:
5141:
5132:
5130:
5126:
5124:
5101:
5089:
5085:
5079:
5075:
5071:
5064:
5060:
5054:
5050:
5046:
5043:
5036:
5033:
5029:
5025:
5022:
5018:
5009:
5006:
5002:
4996:
4992:
4986:
4982:
4976:
4971:
4964:
4959:
4956:
4952:
4948:
4941:
4937:
4932:
4929:
4922:
4913:
4910:
4906:
4902:
4896:
4891:
4886:
4883:
4877:
4872:
4868:
4860:
4843:
4838:
4835:
4831:
4827:
4823:
4814:
4811:
4807:
4803:
4796:
4792:
4788:
4782:
4777:
4773:
4765:
4764:
4763:
4747:
4743:
4739:
4734:
4730:
4726:
4716:
4702:
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4382:semiconductor
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4366:nucleic acids
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2014:
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1972:
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1014:
1010:
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969:
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960:
950:
949:10 coulombs.
944:
924:
919:
915:
906:
903:
900:
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894:
888:
885:
882:
876:
872:
867:
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801:
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538:
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517:
506:
498:
494:
490:
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480:
475:
470:
466:
462:
457:
453:
444:
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432:
425:
408:
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396:
393:
375:
371:
363:
346:
342:
334:
333:
332:
318:
310:
306:
300:
296:
289:
285:
279:
276:
268:
264:
255:
250:
239:
231:
227:
218:
213:
202:
194:
190:
181:
176:
165:
162:
157:
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141:
132:
128:
126:
116:
113:
110:
109:
105:
102:
99:
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91:
88:
87:
83:
80:
77:
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5736:
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5583:
5561:(1): 45–49.
5558:
5554:
5548:
5515:
5511:
5505:
5496:
5491:
5467:. Retrieved
5464:web.nmsu.edu
5463:
5454:
5442:. Retrieved
5414:. Retrieved
5386:. Retrieved
5354:
5284:
5280:
5251:
5244:
5193:
5189:
5183:
5170:Double layer
5159:
5155:
5151:permittivity
5147:
5133:Limitations
5127:
5123:Debye length
5117:
4717:
4379:
4359:
4279:
3764:
3761:
3547:functional:
3541:
3440:
3416:
3404:
3400:
3387:
3370:Debye length
3366:
3296:
3046:
2973:
2748:
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2476:
2295:
2233:
2122:
2092:
2034:
2001:
1955:
1868:
1838:Debye length
1751:
1689:
1678:denotes the
1526:
1408:
1214:
1210:
1194:
1169:
1161:
951:
942:
665:
436:
137:
122:
100:Gouy-Chapman
84:Assumptions
66:double layer
51:
31:
29:
5632:(1): 1–16.
3423:erythrocyte
1201:numerically
125:capacitance
5915:Categories
5469:2018-06-01
5176:References
3427:glycocalyx
3363:Conditions
2239:parameter
1866:equation.
1207:Geometries
37:mean-field
5803:250813154
5540:120529487
5236:119468015
5228:1292-8941
5086:ε
5076:ε
5037:−
5026:−
5003:λ
4993:ε
4983:ε
4953:λ
4949:−
4938:τ
4930:−
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4892:π
4878:≈
4869:ρ
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4320:Δ
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3805:Δ
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3705:ε
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3270:−
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3253:−
3222:−
3176:−
3168:⋅
3159:−
3091:
3085:⋅
3054:ψ
3032:ψ
2994:ψ
2985:≡
2947:−
2939:⋅
2930:−
2899:−
2853:−
2845:⋅
2836:−
2768:
2719:−
2711:⋅
2702:−
2671:−
2625:−
2617:⋅
2608:−
2454:−
2348:−
2308:−
2259:ψ
2250:≡
2184:∑
2157:ε
2153:ε
2043:λ
2015:∘
1983:−
1964:λ
1917:ε
1913:ε
1848:λ
1812:≤
1809:ψ
1774:≪
1767:ψ
1723:−
1709:ψ
1702:ψ
1692:potential
1609:π
1547:Φ
1535:ψ
1511:ψ
1508:−
1500:δ
1497:−
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1470:ψ
1430:ψ
1360:ψ
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1293:⋅
1281:ε
1277:ε
1235:ψ
1100:ψ
1087:−
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1038:−
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998:−
990:−
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889:ψ
883:−
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790:ψ
777:⋅
759:−
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728:−
720:−
695:ψ
491:−
481:⋅
405:ε
372:ε
343:ρ
307:ε
297:ε
286:ρ
280:−
261:∂
256:ψ
247:∂
224:∂
219:ψ
210:∂
187:∂
182:ψ
173:∂
163:ψ
154:∇
89:Helmholtz
5856:18058376
5714:18432617
5309:17184352
5301:12501158
5164:See also
4703:′
4656:′
4606:′
4362:proteins
3503:′
3488:′
3431:spectrin
941:, where
5836:Bibcode
5783:Bibcode
5741:Bibcode
5705:2599918
5664:9876118
5655:1302495
5634:Bibcode
5563:Bibcode
5520:Bibcode
5444:June 1,
5416:June 1,
5388:June 1,
5208:Bibcode
1648:(where
660:kelvins
632:is the
429:is the
43:Origins
5895:AFMPB
5884:MIBPB
5854:
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4762:where
3859:where
3516:where
3421:of an
2093:where
537:where
441:. The
331:where
78:Theory
5852:S2CID
5826:arXiv
5799:S2CID
5536:S2CID
5439:(PDF)
5411:(PDF)
5305:S2CID
5232:S2CID
5198:arXiv
2058:0.304
1179:; in
111:Stern
5710:PMID
5660:PMID
5594:ISBN
5446:2014
5418:2014
5390:2014
5359:ISBN
5297:PMID
5257:ISBN
5224:ISSN
5139:edit
4400:and
4344:solv
4150:solv
3845:solv
3429:and
3411:tRNA
842:and
707:and
56:and
30:The
5878:Zap
5844:doi
5791:doi
5749:doi
5700:PMC
5692:doi
5650:PMC
5642:doi
5571:doi
5559:258
5528:doi
5289:doi
5216:doi
4858:and
4328:mob
4296:out
4210:exp
4048:mob
3829:mob
3628:exp
1183:as
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3314:=
3309:0
3305:y
3278:x
3274:K
3266:e
3259:)
3256:1
3248:2
3244:/
3238:0
3234:y
3229:e
3225:(
3219:1
3216:+
3211:2
3207:/
3201:0
3197:y
3192:e
3184:x
3180:K
3172:e
3165:)
3162:1
3154:2
3150:/
3144:0
3140:y
3135:e
3131:(
3128:+
3125:1
3122:+
3117:2
3113:/
3107:0
3103:y
3098:e
3080:e
3076:T
3071:B
3067:k
3063:2
3057:=
3009:T
3004:B
3000:k
2991:e
2982:y
2955:x
2951:K
2943:e
2936:)
2933:1
2925:2
2921:/
2915:0
2911:y
2906:e
2902:(
2896:1
2893:+
2888:2
2884:/
2878:0
2874:y
2869:e
2861:x
2857:K
2849:e
2842:)
2839:1
2831:2
2827:/
2821:0
2817:y
2812:e
2808:(
2805:+
2802:1
2799:+
2794:2
2790:/
2784:0
2780:y
2775:e
2762:2
2759:=
2756:y
2727:x
2723:K
2715:e
2708:)
2705:1
2697:2
2693:/
2687:0
2683:y
2678:e
2674:(
2668:1
2665:+
2660:2
2656:/
2650:0
2646:y
2641:e
2633:x
2629:K
2621:e
2614:)
2611:1
2603:2
2599:/
2593:0
2589:y
2584:e
2580:(
2577:+
2574:1
2571:+
2566:2
2562:/
2556:0
2552:y
2547:e
2540:=
2535:2
2531:/
2527:y
2523:e
2498:2
2494:/
2490:y
2486:e
2460:)
2457:1
2449:2
2445:/
2439:0
2435:y
2430:e
2426:(
2423:)
2420:1
2417:+
2412:2
2408:/
2404:y
2400:e
2396:(
2391:)
2388:1
2385:+
2380:2
2376:/
2370:0
2366:y
2361:e
2357:(
2354:)
2351:1
2343:2
2339:/
2335:y
2331:e
2327:(
2321:=
2316:x
2312:K
2304:e
2274:T
2269:B
2265:k
2256:e
2247:y
2211:2
2204:i
2200:Z
2192:i
2188:c
2178:T
2172:B
2167:k
2161:0
2147:2
2143:e
2135:=
2131:K
2106:0
2102:c
2074:0
2070:c
2064:m
2061:n
2052:=
2047:D
2020:C
1986:1
1978:K
1973:=
1968:D
1938:T
1932:B
1927:k
1921:0
1906:2
1902:e
1896:0
1892:c
1888:2
1881:=
1877:K
1852:D
1822:V
1819:m
1789:T
1783:B
1778:k
1770:|
1764:|
1760:e
1731:x
1727:K
1719:e
1713:0
1705:=
1664:0
1661:e
1657:n
1630:0
1627:e
1623:n
1617:2
1613:e
1606:4
1601:T
1598:k
1591:=
1586:D
1583:e
1579:R
1555:T
1552:k
1544:e
1538:=
1504:e
1488:e
1484:=
1478:r
1475:d
1467:d
1459:r
1456:L
1451:+
1443:2
1439:r
1435:d
1425:2
1421:d
1394:]
1386:T
1380:B
1375:k
1369:)
1366:x
1363:(
1357:e
1348:e
1337:T
1331:B
1326:k
1320:)
1317:x
1314:(
1308:e
1302:e
1297:[
1285:0
1272:e
1267:0
1263:c
1256:=
1248:2
1244:x
1240:d
1230:2
1226:d
1148:.
1144:]
1136:T
1131:B
1127:k
1121:)
1118:z
1115:,
1112:y
1109:,
1106:x
1103:(
1097:e
1091:e
1080:T
1075:B
1071:k
1065:)
1062:z
1059:,
1056:y
1053:,
1050:x
1047:(
1041:e
1032:e
1027:[
1020:e
1015:0
1011:c
1007:=
1003:)
994:c
985:+
981:c
977:(
973:e
970:=
965:e
947:×
943:e
925:T
920:B
916:k
910:)
907:z
904:,
901:y
898:,
895:x
892:(
886:e
877:e
868:0
864:c
860:=
855:+
851:c
826:T
821:B
817:k
811:)
808:z
805:,
802:y
799:,
796:x
793:(
787:e
781:e
772:0
768:c
764:=
755:c
731:e
725:=
716:W
692:e
689:=
684:+
680:W
669:ψ
662:.
646:T
636:,
617:B
612:k
587:i
583:W
558:0
553:i
549:c
525:,
518:T
512:B
507:k
499:i
495:W
485:e
476:0
471:i
467:c
463:=
458:i
454:c
433:.
427:ψ
409:0
376:r
347:e
319:,
311:0
301:r
290:e
277:=
269:2
265:z
251:2
240:+
232:2
228:y
214:2
203:+
195:2
191:x
177:2
166:=
158:2
20:)
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