Discussion Forum

In general the local surface force density on a dielectric boundary is the sum of the up and down Maxwell stress tensor variables (es.unTx+es.dnTx , es.unTy+es.dnTy , es.unTz+es.dnTz). It is really a difference but the "up" and "down" surface normal directions are accounted for in the variable definitions. On a conducting body, the fields in the conductor are zero so one of the "up" and "down" contributions vanish. If you have a license for the MEMS Module, the "Electromechanical Forces" multiphysics coupling will account for these forces and also add any volumetric forces due to gradients in the electric properties in the interior of a domain.

Best regards,

In general the local surface force density on a dielectric boundary is the sum of the up and down Maxwell stress tensor variables (es.unTx+es.dnTx , es.unTy+es.dnTy , es.unTz+es.dnTz). It is really a difference but the "up" and "down" surface normal directions are accounted for in the variable definitions. On a conducting body, the fields in the conductor are zero so one of the "up" and "down" contributions vanish. If you have a license for the MEMS Module, the "Electromechanical Forces" multiphysics coupling will account for these forces and also add any volumetric forces due to gradients in the electric properties in the interior of a domain.

Best regards,

Dear Mr. Olsson, First of all, thank you very much for your good advice. But I still feel uncertain. I want to simulate the electrostatic attraction force through the AC/DC module of COMSOL. The schematic diagram of my 2D model is shown in the attachment (figure.jpg). When the high potential difference is applied between metal electrodes (positive electrode is applied high potential, negative electrode is applied low potential or grounding), the fringe electric field generated by the metal electrodes will produce electrostatic attraction force on the insulating substrate (due to polarization) or conductive substrate (due to electrostatic induction). 1. In the post-processing, I obtain the electrostatic attraction force per unit length by integrating the Y component of Maxwell's surface stress tensor on the upper surface of the substrate (as shown in the red solid line in figure.jpg). However, because the surface is the boundary of two different materials (that is, there are different dielectrics on both sides of the boundary), its normal electric field intensity will be discontinuous, resulting in the discontinuity of the Y component of the corresponding Maxwell’s surface stress tensor. At this point, I'm not sure which Y component of Maxwell's surface tensor (es.unTy, es.dnTy, es.unTey, es.dnTey, es.nTy) in COMSOL is used for integration. And as you said, the local surface force density (Y component of Maxwell's surface stress tensor) on a dielectric boundary should be the sum of the up and down Maxwell stress tensor variables (such as es.unTy+es.dnTy, or es.unTey+es.dnTey). However, I see that some references seek the difference of the up and down Maxwell stress tensor variables rather than the sum of the up and down Maxwell stress tensor variables (such as formula 8-10 in the attachment(1.pdf) and formula 9 in the attachment(1.pdf)). And the local surface force density (Y component of Maxwell's surface stress tensor, such as es.unTy and es.dnTy) on a dielectric boundary obtained by me will be different in symbols due to the difference of substrate materials. For example, when the boundary is the boundary between air and upper surface of substrate and the substrate is insulator (glass), the es.unTy and the es.dnTy are positive, and the es.dnTy is equal to the es.nTy. But when the substrate is conductor (Aluminum), the es.unTy is negative, and the es.dnTy is positive, as well as the es.dnTy is equal to the es.nTy. Furthermore, when the boundary is the boundary between the lower surface of silicone insulator and the upper surface of substrate (that is, there is no air gap between silicon insulator and substrate) and the substrate is conductor (Aluminum), the es.unTy is positive, and the es.dnTy is negative, as well as the es.dnTy is still equal to the es.nTy. I know the “up” and “down” means the surface normal directions through the variable definitions in the help files. But how do I understand the symbolic differences of es.unTy, es.dnTy in different substrate material? 2. When the substrate is conductor (Aluminum) in my model, I don’t find that one of es.unTy or es.dnTy would valish. I agree with you, but I can't explain that both es.unTy and es.dnTy exist. Should I add special boundary conditions to the aluminum substrate, such as grounding it? 3. I know that I can do the same calculation by volume force, which is essentially integrating the Y component of Maxwell's surface tensor in the all outer surface of the medium. However, for two-dimensional simulation, I need to integrate the Y component of Maxwell's surface tensor in the upper surface of the substrate separately. That's why the problems I encountered above appeared.

Thank you very much again. I look forward to your further reply.

Yours sincerely

The source of the difficulties here is that you are trying to model what I would call an electrostatic contact problem. That is when you have a dielectric with embedded sources being attracted to a metallic surface and making contact - similar to a fridge or whiteboard magnet attaching to a ferromagnetic plate. The key to success is to realize that there is a small or virtual air gap between the pieces. That is, when pulling apart, an air gap wll form rather than a gap of metal or dielectric. You have two choices to do this illustrated by the two attached models. One is intrducing a small gap in the geometry that is meshed using a a mapped mesh to avoid an excessive number of elements. The other model uses a virtual air gap using the Thin Low Permittivity gap feature. In both cases, the metal needs to be grounded. And, as the Thin Low Permittivity gap feature is only allowed on interior boundaries, a domain constraint has to be applied for the grounded electrode. Note that the attaching force is obtained as a limiting value as you make the air gap thinner and thinner, hence the parametric studies.

When it comes to using the up or down Maxwell stress tensors and difference or sum. One should use a sum or, on an outer boundary, the down variant (as then the up variant does not exist). The two variants are computed using antparallel surface normals so the "minus sign" is inherently accounted for - hence using the sum. Note that when making an internal boundary external by disabling an adjacent domain, the definitions of "up" and "down" may switch - you can see that when comparing expressions used in the Line Integration features in the Results section for the two models - one has an internal ground bundary whereas the other one has an external ground boundary and I chose to integrate the force on the electrode rather than on the dielectric-air interface.

In the models, I assumed that there is a larger array of electrodes so I used periodic conditions for a symmetric unit cell. Actually a smaller antisymmetric unit cell could have been used.

Best regards, Magnus

Dear Mr. Olsson,

Thank you again for your help and advices. According to what you said, I have a better understanding of my model. What I simulate is the attraction of electrostatic chuck composed of metal electrodes on insulating target surface or conductive target surface. In the schematic diagram, I call the insulating target surface or conductive target surface as the substrate. So my substrate (whether insulated or conductive) is not like the grounding electrode you said. In my model, the negative electrode can be grounded, while the positive electrode applies a high potential. And when my substrate is conductive metal, I prefer to apply suspension potential on its four boundaries. Following your suggestion, I imposed periodic conditions on the left and right boundaries of the model to replace the air solution domain or infinite element domain I imposed earlier. Currently, I still have some questions about what you said. 1. Why is it necessary to add a very small or virtual air gap between dielectric material and metal substrate to simulate the close contact between them? In my model, I consider the real air gap between dielectric and metal substrate, but our air gap is not as small as you preset. When I do not consider the air gap between the dielectric and the metal substrate, I let the dielectric directly contact the metal substrate in the model, that is, I do not preset a very small or virtual air gap between them as you do. But the result of our doing seems not right. Please give me a further explanation. Thanks! 2. In the model file (ES_force.mph) you provided me, I didn't see you set material properties for metal electrodes and metal substrates. Why? And, although you have added material properties (metal) to the metal electrode and metal substrate in the model file (ES_force_1.mph), you have not set the conductivity of the metal under the material properties. Why is that? In my model, the metal electrodes are usually made of copper, and the object to be attracted by the electrostatic chuck is the insulating surface (glass) or conductive surface (aluminum), so I give the metal electrodes the properties of copper (including relative dielectric constant and conductivity) and give the substrate the properties of glass (including relative dielectric constant) or aluminum (including relative dielectric constant and conductivity), respectively. 3. In your model, why do you impose grounding boundary conditions on the upper surface of the metal substrate? 4. Why is the domain you select under the global “force calculation” node the upper half of the substrate (I think of it as electrostatic chuck)? According to the Newton’s Third Law of Motion, the electrostatic attractive force of the electrostatic chuck to the substrate is equal to that of the substrate to the electrostatic chuck. So I think your settings will not affect the results. Although I have also verified this, I prefer to choose the substrate as the domain for the “force calculation”. I'm not sure if that's what you mean. Please criticize and correct me. Thanks! 5. I'm confused by what you said--“One should use a sum or, on an outer boundary, the down variant (as then the up variant does not exist). The two variants are computed using antparallel surface normals so the "minus sign" is inherently accounted for - hence using the sum.” I don't know if your sum means es.unTey + es.dnTey or |es.unTey| +|es.dnTey| when both the es.unTey and es.dnTey exist. (|es.unTey| means the absolute of es.unTey and |es.dnTey| means the absolute of es.dnTey.) What I understand is that the signs of es.unTey and es.dnTey are related to the normal component on the outer surface of the object and the global Cartesian coordinate direction of the model. For the es.dnTey, it describes the Y component of Maxwell's surface stress tensor in the domain below the boundary. Because the normal direction of the outer surface of the domain below the boundary is consistent with the Y direction of the global Cartesian coordinates, the sign of es.dnTey is positive. However, for the es.unTey, it describes the Y component of Maxwell's surface stress tensor in the domain above the boundary. Because the normal direction of the outer surface of the domain above the boundary is opposite with the Y direction of the global Cartesian coordinates, the sign of es.unTey is negative. Considering the sign of es.unTey and es.dnTey, if I directly perform this sum operation to add es.unTey and es.dnTey (namely, es.unTey + es.dnTey), it actually means that |es.dnTey|-| es.unTey|. So I’m not sure whether I should execute es.unTey+es.dnTey or |es.unTey|+|es.dnTey|. In addition, what is the actual physical meaning of the signs of es.unTey and es.dnTey? Can I understand that: the “+” sign means attraction and the “-” sign means repulsion? If so, does that mean I should execute es.unTey + es.dnTey? Thank you again for your detailed answers. I look forward to your further reply. Thanks!

The key point is understanding that the origin of the electrostatic forces in a device like this is the high electric energy density in the air gap that forms between the chuck and the metallic plate when you separate those. Thus, the Maxwell stress tensor must be evaluated in that (real or virtual) air gap. If you were to immerse the entire device in a dielectric liquid with the same properties as the solid dielectric (or a conducting liquid), there would not be much attractive (or repulsive) force. This is a bit easier to understand if you apply the principle or virtual work, see: https://en.wikipedia.org/wiki/Virtual_work That air gap energy density is generated by the work performed by the applied separating force. The energy density would be zero in a gap filled by liquid metal and small in a gap filled by high permittivity material. If you evaluate the Maxwell stress inside the metal or inside the dielectric, it correponds to those (incorrect) cases.

If you look at the equation solved in electrostatics, the conductivity does not enter at all and metallic domains are anyway equipotential domains. Thus metallic domains are typically not part of the selection for the electrostatics physics but that requires an active decision/action by the user - just setting the domain material to something like copper does not work (no built-in logics for that as it could potentially misfire badly). On the surface of the (excluded) electrode domains, one must set, either a fixed potential (could be ground) or a floating potential. In my model, the metallic plane can be set to floating potential, see attached version but its potential will be close to ground as the active electrodes are set to positive and negative values symmetrically around zero. Just keep in mind that, in many devices, large metal planes are often physically or effectively grounded (the latter by large parasitic capacitances and resistances).

How to go about modeling capacitve and conductive devices actually depends on the time scales involved as dielectrics typically have nonzero but small conductivity. You may consult the AC/DC Module User’s Guide documentation on "Charge Relaxation Theory" for a better understanding. Depending on time scales involved materials may behave as conductors or insulators.

Whether you evaluate the Y force on the chuck or ground electrode should not make any difference as you say - except for the sign of course. In some situations, like when pieces are far apart and the fields around one piece are dominated by precsribed sources (like for a permanent magnet), limited numeric precision may cause incorrect results for the force on that piece. That is not a problem here as the gap size approaches zero.

Lastly 'es.unTey + es.dnTey', is the expression to be used for the Y surface force. As signs typically are the consequences of bookkeeping decisions (here the choice of different local reference directions for the "up" and "down" variables) one cannot figure out or justify a "correct choice" from physics only.

Best regards

Dear Mr. Olsson,

Thank you very much for your prompt reply！ 1、I have learned from the literatures that the electrostatic attractive force of electrostatic chuck on insulating substrate comes from the electric polarization of insulating substrate surface under the electric field generated by electrostatic chuck; The electrostatic attractive force of electrostatic chuck on conductive substrate comes from the induced charge on the surface of conductive substrate under the electric field generated by electrostatic chuck. I'm not sure whether the above two mechanisms of electrostatic attractive force are consistent with what you said. Please have a further explanation. Thanks! 2、I can be sure that the solution of Laplace equation does not involve conductivity in electrostatic field. So what I understand is that whether the metal electrodes and metal substrate are endowed with material properties (conductivity) in my model has no impact on my concern. Is that so? 3、According to your tips, when I simulate the attractive force of electrostatic chuck on conductive substrate, I can apply boundary conditions as follows: the positive electrode is applied with high potential, the negative electrode is grounded, and the four edges of metal substrate apply suspension potential. And when I simulate the attractive force of electrostatic chuck on insulating substrate, I just need to apply a high potential to the positive electrode and the negative electrode is grounded. Is that so? 4、Can the signs of es.unTey and es.dnTey be understood as attraction and repulsion? For example, if the sign of es.unTey + es.dnTey is negative, does this mean that the electrostatic force between the electrostatic chuck and the substrate is repulsive rather than attractive? Thank you again for your detailed answers. I look forward to your further reply. Thanks!

1) Computing the force assuming an infinitesimal air gap is the correct way unless the surfaces are extremely smooth! The force would immediately jump to the higher "air gap value" as separation starts. From any practical standpoint that is the value you seek! However, if the surfaces are extremely flat and smooth, there are all sorts of adhesive effects that come into play and those can typically not be explained by electrostatics only. Then you have to consider complicated surface physics/chemistry so trying to find an electrostatic force for the case with perfectly smooth, planar surfaces in contact is an irrelevant philosophical excercise! You would definitely want avoid that case when designing a device of this type as it may get stuck forever on the substrate due to adhesion. The whole idea is that the attractive force should be of purely electrostatic nature and controllable.

2) Yes, a material may be equipped with a lot of properties that are irrelevant for a particular application. COMSOL will throw an error if a required material property is missing but we do not warn for unused/irrelevant material properties as that would litter the user interface and rightly be considered as desinformation.

3) Yes, but keep in mind that, in real life, any large electrode is often effectively grounded due to its high stray conductance (and capacitance) to an external ground. Thus a sound engineering design is probably to put the energized (smaller) electrodes at +/- voltages symmetrically around ground and to ground the metallic substrate. For the force computation, it should not matter but if you design a device with a huge metalic substrate it may not be realistic to assume that it is at a truly floating potential that is only affected by capacitive coupling to the energized electrodes. Other (stray) couplings can easily be stronger.

4) "Repulsive" or "attractive" depends on the direction to the "other piece" so that is a matter of (your) bookkeeping. If the other piece is in the positive Y direction, it is attractive - otherwise it is repulsive.

Dear Mr. Olsson, Thank you very much for your further explanation! With your help, I can basically understand all the questions I consulted you above. But as for the signs of es.unTey and es.dnTey, I'd like to consult you in combination with the schematic diagram (as shown in figure.jpg in the attachment). Because I don't quite understand what you mean by "other piece" at last. I'm afraid I misunderstood it. In the figure.jpg, the Cartesian coordinates are represented by x and y, respectively. The boundary between the air gap and the substrate is described by the thick red solid line. The Nsubstrate represents the normal vector of the substrate at the boundary. And the Nair represents the normal vector of the air gap at the boundary. The results of the solution is es.unTey < 0, es.dnTey >0 (As you get from your model). Now, my understanding is that: es.dnTey >0, and its normal vector direction coincides with the positive direction of Cartesian coordinates y, so the “+” sign of es.dnTey means “attractive”; similarly, es.unTey < 0, and its normal vector direction is opposite with the positive direction of Cartesian coordinates y, so the “-” sign of es.unTey means “repulsive”. Is that what you mean? Thank you again for your patient explanation. I look forward to your further reply. Thanks!

The surface Maxwell force density is defined as (es.dnTex+es.unTex , es.dnTey+es.unTey). In the attached version of the model, this is plotted on the surfaces of the substrate (white arrows) and the chuck (black arrows). Now, when looking at the sign of the Y component, it is clear that interpreting that as "attraction" or "repulsion" depends on what piece (substrate or chuck) you are evaluating at. Further it depends on the relative positions of the chuck and substrate - rotate the geometry by 180 degrees and the signs/vector directions are reversed, rotate by 90 degrees and you have to look at the X component, etc. etc. That is what I mean by "your bookkeeping" and why I am reluctant to state something like "positive sign means attraction" as it depends on how the model is defined/oriented and on what side of the air gap you are evaluating.

Best regards

Dear Mr. Olsson, Thank you very much for your patient explanation and help. Based on your explanation, I fully understand the problem now. Thanks!

The surface Maxwell force density is defined as (es.dnTex+es.unTex , es.dnTey+es.unTey). In the attached version of the model, this is plotted on the surfaces of the substrate (white arrows) and the chuck (black arrows). Now, when looking at the sign of the Y component, it is clear that interpreting that as "attraction" or "repulsion" depends on what piece (substrate or chuck) you are evaluating at. Further it depends on the relative positions of the chuck and substrate - rotate the geometry by 180 degrees and the signs/vector directions are reversed, rotate by 90 degrees and you have to look at the X component, etc. etc. That is what I mean by "your bookkeeping" and why I am reluctant to state something like "positive sign means attraction" as it depends on how the model is defined/oriented and on what side of the air gap you are evaluating.

Best regards

Dear Mr. Olsson, I'm sorry to bother you again. Continuing with the previous question, I would like to consult you again about how to realize "zero potential at infinity" in COMSOL. In my previous model, the substrate has finite thickness, and I directly grounded the lower boundary of the substrate. Now, I want to simulate a more realistic situation, that is, the substrate has infinite thickness (in the - Y direction) compared with the electrostatic chuck. In this case, how can I add boundary conditions to the substrate? I have tried to use the substrate as an infinite element domain directly, or add an additional area under the substrate and set it as the infinite element domain, and then ground the bottom boundary of the infinite element domain. I think that after this, the change of substrate thickness will no longer affect the electrostatic attractive force on the upper surface of the substrate, but the actual result is that the electrostatic attractive force is still dependent on the thickness of substrate. If it is assumed that the substrate has infinite thickness and the potential at infinity is zero, will the actual thickness of substrate in the model affect the electrostatic attractive force on the upper surface of the substrate? At this point, I would like to consult you again: in the examples you gave me, you grounded the upper surface of the substrate. But in practice, the grounding of the lower surface of the substrate is more consistent with the actual situation. Why didn't you ground the lower surface of the substrate? However, I found that only when the top of the substrate is grounded, the electrostatic attractive force is not dependent on the thickness of substrate. I look forward to your reply. Thanks!

I guess your model is in planar 2D, right? Then, the assumption of zero potential at infinity is not correct. It is inherent in the assumption in planar 2D that we are modeling an object that is of infinite extent in the out-plane direction. For example two concentric circular electrodes with air in between (infinitely long cylindrical capacitor) with the inner electrode holding a fixed charge, one can easlily show that the electric field must scale as 1/radius. However, if you try to intergrate that electric field to infinite radius (to get the capacitor voltage), that integral diverges (as log(radius)). Thus, in 2D, the best you can do is to assume gound at finite distance.

Note also that the problem arises only when you are considering a 2D monopole field. If you have two or more poles (electrodes) separated by a finite distance and held at different potentials (or with balancing charges, i.e. no monopole component), you will have a potential that approaches a constant value at infinity. This is because dipole fields (and higher multipole fields) fall off more rapidly with distance.

I should also add that in my model, a metallic substrate was assumed and that is by definition an equpotential domain in electrostatics (no currents and, hence, no resistive voltage drop). Thus, if it is grounded at any end, all of it will be ground.

Best regards

Dear Mr. Olsson, * Yes, I did do a two-dimensional simulation. * Maybe you misunderstood me. As shown in the figure.jpg in the attachment, my means is that the substrate has a great thickness relative to the electrostatic chuck in the two-dimensional plane, that is the lower surface of the substrate is infinite in the negative direction of y (-y). How to set in COMSOL so that the substrate has infinite thickness and the potential of its lower surface at infinity is zero? * In my model, I apply a positive voltage of 2500V to the positive electrode and a negative voltage of -2500V to the negative electrode. When the substrate is the metallic (such as aluminum), I need to set the dielectric constant in the metal material properties as large as possible (such as 50000) in order to make the grounding results of the upper or lower surface of the metal substrate consistent. That's what you said--“I should also add that in my model, a metallic substrate was assumed and that is by definition an equpotential domain in electrostatics (no currents and, hence, no resistive voltage drop). Thus, if it is grounded at any end, all of it will be ground.” * However, if I follow the default metal material properties of the software (dielectric constant is 1), the results of the upper surface grounding of the metal substrate and its lower surface grounding is different. However, if I apply the floating potential to the upper surface of the metal substrate and ground the lower surface, the result of which is consistent with that of only grounding the upper surface of substrate. And it is also consistent with the result that the dielectric constant of metal is 50000. * Now I don't quite understand how to set the boundary conditions if the metal substrate is the insulator (such as glass) ? I have grounded the upper or lower surface of the insulating substrate, and the results are different. And only when the upper surface of the insulating substrate is grounded, the result is not affected by the thickness of substrate. * I look forward to your reply. Thanks!

First of all, I would like to repeat that the Electrostatics formulation/interface is not intended for the simulation of the interior of metallic domains! Only dielectric properties are properly treated. Applying a metallic material to a domain will not result in that domain becoming an isopotential domain. It is up to you to exclude metallic domains from the Electrostatics selection and apply suitable boundary conditions on the boundaries of the excluded domains.

In your case, grounding the upper surface of the substrate (and excluding its domain) only makes sense if the substrate is metallic. If the substrate is a non-conducting dielectric, you should include its domain and apply either Ground or Zero Charge to its lower surface. If it has a high permittivity and is thick enough, the forces should not depend much on which of those you choose, see attached model and try enabling/disabling the Ground feature. There is not much of a difference in the force values.

I would say that in planar 2D, one should never apply Ground on the outside of an infinite elements domain for the reasons I explained before (monopole potential diverging as you go to infinity).

Dear Mr. Olsson, Thank you very much for your patient explanation and help. I get the idea. Thanks!