3.10: Electric Korteweg-Helmholz Force Density
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The thermodynamic technique used in this section for deducing the electric force density with combined effects of free charge and polarization is a generalization of that used in determining discrete forces in Sec. 3.5. This principle of virtual work is exploited because it is not practical to predict the relationship between microscopic and macroscopic fields.
In any derivation of a force density, it is important to be clear about (a) what empirically determined information is required, and (b) what postulates or assumptions are incorporated into the derivation or are implicit to an application of the force density. Generally, empirically determined information can be used to replace assumptions. As derived here, the only empirical information required is an electrical constitutive law relating the macroscopic electric field to the polarization density \(\overrightarrow{P}\) (or displacement \(\overrightarrow{D}\)). This relationship is typically determined by making electrical measurements on homogeneous samples of the material. These amount to measurements of the terminal characteristics of capacitor-like configurations incorporating samples of the material. (In the lumped-parameter systems of Sec. 3.5, the analogous empirical information was the electrical terminal relation.) With so little empirical information, the force density can only be identified if the system considered is a conservative thermodynamic subsystem. Thus, the force density is derived picturing the system as having no dissipation mechanisms. (The same conservative system is considered in Sec. 3.5 to find discrete forces.) The assumption is then made that the force density remains valid even in modeling systems with dissipation. If dissipation mechanisms were to be incorporated into the system considered,then a virtual power principle could be exploited to find the force density, but additional empirical information would be required.
Experiments show that, for a wide range of materials, electrical constitutive laws take the form of state functions
\[ \overrightarrow{E} = \overrightarrow{E} (\alpha_1...\alpha_m, \overrightarrow{D}) \, or \, \overrightarrow{D} = \overrightarrow{D} (\alpha_1...\alpha_m, \overrightarrow{E}) \label{1} \]
The \(\alpha\)'s are properties of the material. Thus, if measurements are made on a homogeneous sample of the material, the a's are varied by changing the composition of the sample. For example, \(\alpha_1\) might be the concentration of dipoles of a given species, or the concentration of one liquid in another. The number of \(\alpha\)'s used depends on the specific application. Most important for now is the distinction between changing \(\overrightarrow{E}\) in Equation \ref{1} by changing the material and hence changing \(\alpha\)'s, and doing so by changing \(\overrightarrow{D}\). Some special cases of Equation \ref{1} are given in Table 3.7.1.
| Law | Description |
|---|---|
| \[ \overrightarrow{E} = \varepsilon^{-1} (\alpha_1...\alpha_m) \overrightarrow{D} \nonumber \] | Electrically linear and (fields) collinear |
| \[ E_i = s_{ij} (\alpha_1...\alpha_m) \overrightarrow{D}_j \nonumber \] | Electrically linear and anisotropic |
| \[ \overrightarrow{E} = \varepsilon^{-1} (\alpha_1...\alpha_m, \, D^2) \overrightarrow{D} \nonumber \] | Electrically nonlinear and (fields) collinear |
| \[ E_i = s_{ij} (\alpha_1...\alpha_m, \, D_1, D_2,D_3) \overrightarrow{D}_j \nonumber \] | Electrically nonlinear and anisotropic |
The third case of the table might represent a material in which dipoles are in Brownian equilibrium with.a nonpolar liquid. An applied field tend to line up the dipoles and hence give rise to a polarization density and hence to a contribution to \(\overrightarrow{D}\). In terms of two properties \((\alpha_1,\alpha_2)\), a model including the saturation effect, resulting as all dipoles become aligned with the field, might be
\[ \varepsilon = \frac{\alpha_1}{\sqrt{1 + \alpha_2^2 \overrightarrow{E} \cdot \overrightarrow{E}}} + \varepsilon_o \label{2} \]
Built into this example, and the general relation, Equation \ref{1}, is the assumption that the constitutive law is a state function. It does not depend on rates of change, and it is a single-valued function of the variables and hence not dependent on the path followed to arrive at the given state.
The continuum now considered is not homogeneous, in that at any given instant the \(\alpha\)'s can vary from one position to another. Moreover, for the electromechanical subsystem considered, the properties are tied to the material. As the material moves, properties change. For material within a volume of fixed identity,
\[ \int_V \alpha_i dV = constant \label{3} \]
By definition, the volume \(V\) is always composed of the same material. By definition, the \(\alpha\)'s must satisfy Equation \ref{3} when the subsystem is considered to be isolated from other subsystems.
The finite number of mechanical degrees of freedom for the discrete coupling of Sec. 3.5 is now replaced by an infinite number of degrees of freedom. The mechanical continuum, perhaps a fluid, perhaps a solid, is capable of undergoing the vector deformations \(\delta \overrightarrow{\xi}\). These incremental displacements are viewed as small departures from an equilibrium mechanical configuration which is precisely that for which the force density is required.
Since the time derivative of Eq. 3 vanishes, the generalized Leibnitz rule, Eq. 2.6.5, gives
\[ \frac{d}{dt} \int_V \alpha_i dV = \int_V \frac{\partial{\alpha_i}}{\partial{t}} dV + \oint_S \alpha_i \frac{\partial{\overrightarrow{\xi}}}{\partial{t}} \cdot \overrightarrow{n} da = 0 \label{4} \]
where by definition the velocity of the surface \(S\) is equal to that of the material \((\overrightarrow{v}_s \rightarrow \frac{\partial{\overrightarrow{\xi}}}{\partial{t}})\) Gauss' theorem converts the second integral to a volume integral. Although of fixed identity, the volume is arbitrary, and so it follows from Equation \ref{4} that changes in the property ai are linked to the material de-formations by an expression that is equivalent to Equation \ref{3}:
\[ \delta \alpha_i = - \nabla \cdot (\alpha_i \delta \overrightarrow{\xi} ) \label{5} \]
The framework has now been established for stating and exploiting conservation of energy for the electromechanical subsystem. The procedure is familiar from Sec. 3.5. With electrical excitations absent, a system, such as shown in Fig. 2.13.1, is assembled mechanically. Because the force density of electrical origin is by definition zero during the process, no work is required. The system now consists of rigid electrodes for producing part or all of the electrical excitations and a mechanical continuum in the intervening space. This material is described by Equation \ref{1}. With the mechanical deformations fixed \((\delta \overrightarrow{\xi} = 0)\), the electrical excitations are next raised by placing bulk charges at the position of interest in the material and by raising the potentials on the electrodes. The result is a stored electrical energy given by Eq. 2.13.6:
\[ w = \int_V W dV; \quad W = \int_o^{\overrightarrow{D}} \overrightarrow{E} (\alpha_1...\alpha_m, \overrightarrow{D}^{'}) \cdot \delta \overrightarrow{D}^{'} \label{6} \]
Here, \(V\) is the volume occupied by the material and the fields, and hence excluding the electrodes.
Now, with the net charge on each electrode constrained to be constant, consider variations in the energy caused by incremental displacements of the material. A statement of energy conservation accounting for work done on the external mechanical world by the force density of electrical origin is
\[ \int_V [ \delta W + \overrightarrow{F} \cdot \delta \overrightarrow{\xi}] dV = 0 \label{7} \]
There are two consequences of the incremental displacement. First, the mechanical deformation carries the properties with it, as already stated by Equation \ref{5}. Second, there is a redistribution of the free charge. Because the system is conservative, the free charge is constrained to move with the material.The charge within a volume always composed of the same material particles is constant. Thus, Equation \ref{3} also holds with \(\alpha_i \rightarrow \rho_f\), and it follows that an expression similar to Equation \ref{5} can be written for the change in charge density at a given location caused by the material displacement \(\delta \overrightarrow{\xi} \):
\[ \delta \rho_f = - \nabla \cdot (\rho_f \delta \overrightarrow{\xi}) \label{8} \]
It is extremely important to recognize the difference between \(\delta W\) in Equation \ref{7}, and \(\delta W\) in Sec. 2.13.In Equation \ref{7}, the change in energy is caused by material displacements \(\delta \), whereas in Sec. 2.13 it is due to changes in the electrical excitations. The energy \(W\) is assumed to be a state function of the same variables as used to express the constitutive law, Equation \ref{1}. Hence,
\[ \delta W = \sum_{i=1}^{m} \frac{\partial{W}}{\partial{\alpha_i}} \delta \alpha_i + \frac{\partial{W}}{\partial{\overrightarrow{D}}} \cdot \delta \overrightarrow{D} \label{9} \]
where
\[ \frac{\partial{W}}{\partial{\overrightarrow{D}}} \cdot \delta \overrightarrow{D} \equiv \sum_{i=1}^{3} \frac{\partial{W}}{\partial{D_i}} \delta D_i \nonumber \]
With the understanding that the partial derivative is taken with the \(\alpha\)'s held fixed, it follows from Eq.\ref{6} that
\[ \frac{\partial{W}}{\partial{D_j}} = E_j \label{10} \]
Hence, the last term in Equation \ref{9} is written using Equation \ref{10} with \(\overrightarrow{E}\) in turn replaced by \(- \nabla \phi\). Then, integration by parts* gives
\[ \int_V \frac{\partial{W}}{\partial{\overrightarrow{D}}} \cdot \delta \overrightarrow{D} dV = - \oint_S \phi \delta \overrightarrow{D} \cdot \overrightarrow{n} da + \int_V \phi (\nabla \cdot \delta \overrightarrow{D}) dV \label{11} \]
The part of the surface coincident with the electrode surfaces gives a contribution from each electrode equal to the electrode potential multiplied by the change in electrode charge. Because the electrode charges are held fixed while the material is deformed, this integration gives no contribution. The remaining part of the surface integration is sufficiently well removed from the region of interest that the fields have fallen off sufficiently to make a negligible contribution. Thus, the first term on the right vanishes and, because of Gauss' law, Equation \ref{11} becomes
\[ \frac{\partial{W}}{\partial{\overrightarrow{D}}} \cdot \delta \overrightarrow{D} dV = \int \phi \delta \rho_f dV \label{12} \]
It is now possible to write Equation \ref{7} with effects of \(\delta \overrightarrow{\xi}\) represented explicitly. Substitution of Equation \ref{8} into \ref{12} and then Eqs. \ref{12} and \ref{5} into \ref{9}, and finally of Equation \ref{9} into \ref{7}, gives
\[ \int_V [ - \sum_{i=1}^{m} \frac{\partial{W}}{\partial{\alpha_i}} \delta \cdot (\alpha_i \delta \overrightarrow{\xi}) - \phi \nabla \cdot (\rho_f \delta \overrightarrow{\xi}) + \overrightarrow{F} \cdot \delta \overrightarrow{\xi} ] dV = 0 \label{13} \]
With the objective of writing the integrand in the form \((\, ) \cdot \delta \overrightarrow{\xi} \), the first two terms are integrated by parts. Because the surface integrations are either on the rigid electrode surfaces where \(\delta \overrightarrow{\xi} \cdot \overrightarrow{n}\), or at infinity where the fields have decayed to zero, and \(\overrightarrow{E} = -\nabla \phi\), Equation \ref{13} becomes
\[ \int_V [ \sum_{i=1}^{m} \alpha_i \nabla (\frac{\partial{W}}{\partial{\alpha_i}}) - \rho_f \overrightarrow{E} + \overrightarrow{F} ] \cdot \delta \overrightarrow{\xi} dV = 0 \label{14} \]
It is tempting, and in fact correct, to set the integrand of this expression to zero. But the justification is not that the volume \(V\) is arbitrary. To the contrary, the volume \(V\) is a special one enclosing all of the region occupied by the deformable medium and fields. (The volume integration plays the role of a summation over the mechanical variables for the lumped-parameter systems of Sec. 3.5.) The integrand is zero because \(\delta \overrightarrow{\xi} \) (like the lumped-parameter displacements) is an independent variable. The equation must hold for any deformation, including one confined to any region where \(\overrightarrow{F}\) is to be evaluated:
\[ \overrightarrow{F} = \rho_f \overrightarrow{E} - \sum_{i=1}^{m} \alpha_i \nabla (\frac{\partial{W}}{\partial{\alpha_i}}) \label{15} \]
It is most often convenient to write the second term so that it is clear that it consists of a force density concentrated where there are property gradients and the "gradient of a pressure":
\[ \overrightarrow{F} = \rho_f \overrightarrow{E} + \sum_{i=1}^m \frac{\partial{W}}{\partial{\alpha_i}} \nabla \alpha_i - \nabla [ \sum_{i=1}^{m} \alpha_i \frac{\partial{W}}{\partial{\alpha_i}}] \label{16} \]
The implications of Equation \ref{16} and the method of its derivation are appreciated by considering three commonly encountered limiting cases and then writing Equation \ref{16} in such a way that its relation to the Kelvin force density is clear.
Incompressible Media
Deformations are then such that
\[ \nabla \cdot \delta \overrightarrow{\xi} = 0 \label{17} \]
Because \(\delta \overrightarrow{\xi} \cdot \overrightarrow{n} = 0 \) on the rigid electrode surfaces that comprise part of the surface \(S\) enclosing \(V\) in Equation \ref{7}, any pressure function fr that approaches zero with sufficient rapidity at infinity to make the surface integration there negligible will satisfy the relation
\[ \oint_S \pi \delta \overrightarrow{\xi} \cdot \overrightarrow{n} da = \int_V \nabla \cdot (\pi \delta \overrightarrow{\xi}) dV = 0 \label{18} \]
Thus, Equation \ref{14} remains valid even if the volume integral of Equation \ref{18} is added to it. But, for incompressibledeformations as defined with Equation \ref{17}, \(\nabla \(\pi \delta \cdot \overrightarrow{\xi}) = \nabla \pi \cdot \delta \overrightarrow{\xi} \). Thus, the term added to Equation \ref{14}, like those already appearing in its integrand, can be written with \(\delta \overrightarrow{\xi} \) as a factor. It follows that for incompressible de-formations, the gradient of any scalar pressure, \(\pi\), can be added to the force density of Equation \ref{16}. For example, \(\pi\) might be \(\overrightarrow{P} \cdot \overrightarrow{E}\), since this function decays with distance from the system sufficiently rapidly to make the contribution of the surface integration at infinity vanish. On the basis of this apparent arbitrariness in the force density, the following observation is now made for the first time, and will be emphasized again in Chapter 8. Two force densities differing by the gradient of a scalar pressure will give rise to the same incompressible deformations. Physically this is so because in modeling a continuum as incompressible, the pressure becomes a "left-over" variable. It becomes whatever it must be to make Equation \ref{17} valid. Whatever the \(\pi\) added to the force density of electrical origin, w can be absorbed into the "mechanical" pressure of the continuum-force equation.
For incompressible deformations, where the force density is arbitrary to within the gradient of a pressure, the gradient term can be omitted from Equation \ref{16}, which then takes the convenient form
\[ \overrightarrow{F} = \rho_f \overrightarrow{E} + \sum_{i=1}^{m} \frac{\partial{W}}{\partial{\alpha_i}} \nabla \alpha_i \label{19} \]
This expression concentrates the force density where there are property gradients. In a charge-free system composed of regions having uniform properties, the force density is thus confined to inter-faces between regions.
Incompressible and Electrically Linear
For an incompressible material having the constitutive law
\[ \overrightarrow{D} = \varepsilon_o (1 + \chi_e) \overrightarrow{E} = \varepsilon \overrightarrow{E} \label{20} \]
the susceptibility \(\chi_e\) is conserved by a volume of fixed identity. That is, \(\alpha_1\) can be taken as \(\chi_e\) in Equation \ref{3} and \(m = 1\). Then, from Equation \ref{6},
\[ W = \frac{1}{2} \frac{D^2}{\varepsilon_o (1 + \chi_e)} ; \quad \frac{\partial{W}}{\partial{\chi_e}} = - \frac{\varepsilon_o}{2} E^2 \label{21} \]
and because \(\nabla \chi_e = \nabla [ (1 + \chi_e) ]\), it follows that the force density of Equation \ref{19} specializes to
\[ \overrightarrow{F} = \rho_f \overrightarrow{E} - \frac{1}{2} E^2 \nabla \varepsilon \label{22} \]
Electrically Linear with Polarization Dependent on Mass Density Alone: Certainly a possible parameter \(\alpha_1\) is the mass density p, since then Equation \ref{3} is satisfied. For a compressible medium it is possible that the susceptibility \(\chi_e\) in Equation \ref{20} is only a function of \(\rho\). Then,
\[ \alpha_1 = \rho, \quad \chi_e = \chi_e (\rho), \quad W = \frac{1}{2} \frac{D^2}{\varepsilon_o [1 + \chi_e (\rho)]}; \quad \frac{\partial{W}}{\partial{\rho}} = - \frac{\varepsilon}{2} E^2 \frac{\partial{\chi_e}}{\partial{\rho}} \label{23} \]
and, because \((\partial{\varepsilon} / \partial{\rho}) \nabla \rho = \nabla \varepsilon\), the force density given by Equation \ref{16} becomes
\[ \overrightarrow{F} = \rho_f \overrightarrow{E} - \frac{1}{2} E^2 \nabla \varepsilon + \nabla [\frac{1}{2} \rho \frac{\partial{\varepsilon}}{\partial{\rho}} E^2] \label{24} \]
Because the last term is associated with volumetric changes in the material, it is called the electrostriction force density.
Relation to the Kelvin Force Density: Because \(W = W(\alpha_1,\alpha_2...\alpha_m, )\), the kth component of the gradient of \(W\) is
\[ (\nabla W)_k = \sum_{i=1}^{m} \frac{\partial{W}}{\partial{\alpha_i}} \frac{\partial{\alpha_i}}{\partial{x_k}} + \frac{\partial{W}}{\partial{D_j}} \frac{\partial{D_j}}{\partial{x_k}} \label{25} \]
In view of Equation \ref{10}, it follows that
\[ \sum_{i=1}^{m} \frac{\partial{W}}{\partial{\alpha_i}} \frac{\partial{\alpha_i}}{\partial{x_k}} = \frac{\partial{W}}{\partial{x_k}} - \frac{\partial{}}{\partial{x_k}} (\overrightarrow{E} \cdot \overrightarrow{D}) + D_j \frac{\partial{E_j}}{\partial{x_k}} \label{26} \]
This expression can be substituted for the second term in Equation \ref{16}, which with some manipulation then becomes
\[ \overrightarrow{F} = \rho_f \overrightarrow{E} + \overrightarrow{P} \cdot \nabla \overrightarrow{E} + \nabla [ \frac{1}{2} \varepsilon_o \overrightarrow{E} \cdot \overrightarrow{E} + W - \overrightarrow{E} \cdot \overrightarrow{D} - \sum_{i=1}^{m} \alpha_i \frac{\partial{W}}{\partial{\alpha_i}} ] \label{27} \]
In this form, the force density is the sum of a free charge force density, the Kelvin force density(Eq. 3.6.5) and the gradient of a pressure. This last term can consistently be ignored in predicting the deformations of an incompressible continuum. For such situations, the Kelvin force density or the Korteweg-Helmholtz force density in the form of Equation \ref{19} will give rise to the same deformations. Note that they have very different distributions.
Apparently the last term in Equation \ref{27} represents the interaction between dipoles omitted from the derivation of the Kelvin force density. In fact, this term vanishes when the constitutive law takes a form consistent with the polarization being due to noninteracting dipoles. In that case, the= susceptibility should be linear in the mass density so that Xe cp, where c is a constant. In Equation \ref{23} ,\(\partial{\chi_e}/\partial{\rho} = c\), and evaluation shows that, indeed, the last term in Equation \ref{27} does vanish.
*Integration by parts in three dimensions amounts to
\[ \int_V \psi \nabla \cdot \overrightarrow{A} dV = \int_V \nabla \cdot (\psi \overrightarrow{A}) dV - \int_V \overrightarrow{A} \cdot \nabla \psi dV = \oint_S \psi \overrightarrow{A} \cdot \overrightarrow{n} da - \int_V \overrightarrow{A} \cdot \nabla \psi dV \nonumber \]