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Theory of the Maxwell Pressure Tensor and the Tension in a Water Bridge A. Widom, J. Swain, and J. Silverberg Physics Department, Northeastern University, Boston MA 02115 S. Sivasubramanian NSF Center for High-rate Nanomanufacturing, Northeastern University, Boston MA 02115 Y.N. Srivastava Physics Department & INFN, University of Perugia, Perugia IT 9 0 A water bridgerefers toan experimental“flexiblecable” madeup of puredeionized waterwhich 0 can hang across two supports maintained with a sufficiently large voltage difference. The resulting 2 electricfieldswithin thedeionized waterflexiblecable,maintain atension whichsustainsthewater against the downward force of gravity. A detailed calculation of the water bridge tension will be n provided in terms of the Maxwell pressure tensor in a dielectric fluid medium. General properties a J of the dielectric liquid pressure tensor are discussed along with unusual features of dielectric fluid Bernoulliflowsinanelectricfield. AnalogiesbetweendielectricfluidBernoulliflowsinstrongelectric 7 fieldsand quantumBernoulli flows in superfluidsare explored. 2 ] PACSnumbers: 92.40.Bc,03.50.De r e h I. INTRODUCTION electric field is continuous, the electric field inside the t o cylinderisthesameastheelectricfieldoutsidethecylin- at. Recent observations[1, 2, 3] have been made of water der. On the other hand, the displacement field D = εE is discontinuous atthe endpoints of the cylinder; i.e. the m bridges stretched across supports which are maintained ends ofthe cylinderhaveachargedensity of±σ wherein at large voltage differences. A water bridge is a “flexible - d cable” made up of pure deionized water which has an Q n electric field E in virtue of an applied voltage across the 4πσ =∆D =(ε−1)E =4π . (1) o A supports at the ends of the fluid cable. Previously to c this work, it was not fully understood what forces hold [ Theendofthecylinderwhichisatthetailoftheelectric up the water bridge against the force of gravity. It will field vector has charge +Q while the end of the cylinder 1 here be shown that the forces responsible for holding up whichisatthearrowoftheelectricfieldvectorhascharge v the water bridge follow from the Maxwell electric field −Q. The tension in the cylinder is then evidently given 4 pressure tensor in dielectric polar fluids. In particular, 7 by τ =QE which in virtue of Eq.(1) reads thewaterbridgeviewedasaflexiblecablehasanelectric 2 4 fieldinducedtensionsufficientlylargesoastoexplainthe τ ε−1 2 . waterbridgesupport. Indiscussingthegeneraltheoryof = E . (2) 1 A (cid:20) 4π (cid:21) the pressure tensor for isotropic dielectric polar liquids, 0 9 suchaswater,hydrostaticsandadiabatichydrodynamics In what follows, the stress Eq.(2) will be rigorously 0 shall also be explored. derived from the Maxwell pressure tensor within the v: In Sec.II the thermodynamic laws applied to a fluid cylinder. The ratio of the tension to cylinder weight, i polar dielectric are described in detail. It is shown that Mg =ρALg, obeys X inthepresenceofanelectricfield,therearetwodifferent ar thermodynamic pressures, P and P˜. These turn out to τ (ε−1)E2 be eigenvalues of the full pressure tensor as discussed = , (3) Mg (cid:20) 4πρgL (cid:21) in Sec.III. If an infinitesimal surface area δA has a ⊥ normal perpendicular to the electric field lines, then the whereinρandLrepresent,respectively,themassdensity pressure force is PδA . If an infinitesimal surface area ⊥ and length of the cylinder. Although we have employed δA has a normal parallel to the electric field lines, then k free end boundary conditions to the cylinder, the ten- the pressureforceisP˜δAk. Itis shownthatthese results sion τ is a local stress quantity independent of global completely characterize the pressure tensor. boundary conditions. In Sec.V, we exhibit a plot of the In Sec.IV, we compute the tension in a fluid dielec- hanging water bridge flexible cable for experimental val- tric cylinder. The physics of that calculation is easily ues of the parameters in Eq.(3) and find the agreement explained. Consider a simple cylinder of length L and between theory and experiment to be satisfactory. 2 cross sectional area A = πR . Suppose that a spatially In Sec.VI, the hydrostatics of the dielectric fluid in an uniform electric field E exists in a direction parallel to electric field is explored. The crucial quantity of interest the cylinder axis. Since the tangential component of the is the mean isothermal polarization per molecule which 2 obeys state are non-linear, i.e. isotropy yields a free energy of the form ∂ε p=αTE ⇒ 4παT =m(cid:18)∂ρ(cid:19) , (4) f(ρ,T,E)≡f(ρ,T,E2) (10) T,E so that Eqs.(7) and (10) imply whereinm is the mass ofa single molecule andα is the T isothermalpolarizability. Forthe water liquid andvapor D=εE, (11) phases, the molecular polarizabilities, respectively, obey αgas ≪ αliquid. Fluid dielectric films in strong electric whereinε(ρ,T,E2)=−8π∂f(ρ,T,E2)/∂(E2). Thus,the T T fieldsadsorbedoninsulatingwallstendtoswelltoalarge two possible fluid pressures in Eq.(9) obey thickness. In Sec.VII, the theory of dielectric polar fluid ε Bernoulli flows in strong electric fields is discussed. To- P˜ =P − E2. (12) 4π getherwiththeeffectoffilmthickening,itturnsoutthat the filmiscapableofcrawlingupaninsulatingcontainer It may at first glance appear strange that there are two wall against the force of gravity and flow as in a siphon physicallydifferentthermodynamicpressuresinadielec- over the top of the wall and down the other side. In the tric fluid subject to an electric field. However,the situa- concluding Sec.VIII, the role of Bernoulli flows in form- tion may be clarified when it is realized that due to the ing water bridges will be discussed. Analogies between electric field, the pressure is in reality a tensor. dielectric polar liquid Bernoulli flows in strong electric fields and quantum Bernoulli flows in superfluids will be III. PRESSURE TENSOR explored. These analogies include the ability of films to climb walls against gravitationalforces and even to pass over the tops of these walls over to the other side. In order to compute the Maxwell pressure tensor in a fluid dielectric, imagine that the fluid undergoesa strain whereinafluidparticleatpointrissenttothenewpoint II. THERMODYNAMIC ARGUMENTS r′. Suchatransformationinducesastrainedlengthscale ds2 =dr′·dr′ described by a metric tensor Letf˜(ρ,T,D)representthe Helmholtz free energyper ds2 =g (r)dridrj. (13) ij unit volume for a dielectric fluid of mass density ρ, tem- perature T and Maxwell displacement field D; The pressure tensor P is then describedin terms of the ij free energy change due to a metric strain 1 df˜=−sdT +ζdρ+ E·dD, (5) 4π δF = 1 P δgijdV. (14) ij 2Z wherein s is the entropy per unit volume, ζ is the chemical potential per unit mass and E is the electric field. The free energy variation is described by The thermodynamic pressure which follows from Eq.(5), δF =δ fdV = δfdV + fδdV. (15) P˜ =ζρ−f˜, Z Z Z 1 dP˜ =sdT +ρdζ− E·dD. (6) In detail, the volume element of the strained fluid is de- 4π termined by g(r)=det[g (r)] via ij On the other hand, one may employ the free energy 1 dV = g(r)d3r ⇒ δdV =− g δgijdV. (16) ij 1 2 f =f˜− E·D, p 4π The volume variational Eq.(16) together with mass con- 1 servation in turn implies a mass density variation df =−sdT +ζdρ− D·dE, (7) 4π 1 δρ= g δgijρ. (17) yielding the pressure 2 ij P =ζρ−f, Furthermore,thechangeinthe magnitudeofthe electric 1 field is dP =sdT +ρdζ+ D·dE. (8) 4π δE2 =δgijE E . (18) i j The two different pressures obey We then employ 1 P =P˜+ E·D. (9) ∂f ∂f 2 4π δf = δρ+ δE , (cid:18)∂ρ(cid:19) (cid:18)∂E2(cid:19) T,E2 T,ρ Fora dielectric fluidin anelectricfield, isotropydictates ǫ 2 that D be parallel to E even if the detailed equations of δf =ζδρ− δE . (19) 8π 3 In virtue of Eqs.(15)-(19), 0 1 ε δF = δgij g (ζρ−f)− E E dV. (20) -0.2 ij i j 2Z h 4π i -0.4 From Eqs(8), (14) and (20) we have the final form[4, 5] m) c Hanging Water Bridge Cable for the pressure tensor y / (-0.6 ε P =Pg − E E . (21) ij ij i j 4π -0.8 In dyadic notation, the pressure tensor is given by P=P1− ε EE, -10 0.5 1 (x / cm)1.5 2 2.5 4π εE2 E P=P1− nn with n= , FIG. 1: Employing the experimental example[1, 2] wherein 4π E thesupportsareseparatedby2.5cmandwithρ=1gm/cm3, P=P1+(P˜−P)nn, (22) g = 980 cm/sec2, ε = 80 and E = 33 Gauss, we plot the hanging water bridge flexible cable. The maximum hanging wherein Eq.(12) has been invoked. dip at the center of the water bridge is h = 0.1 cm. The Itisnowclearastowhytherearetwothermodynamic agreement between experiment and the theoretical Eqs.(3) pressures,P andP˜,inSec.II. Foraninfinitesimalsurface and (28) is satisfactory. area δA whose normal is perpendicular to the electric ⊥ field lines, the pressure force is PδA . For an infinites- ⊥ V. HANGING FLEXIBLE CABLE imal surface area δA whose normal is parallel to the k electric field lines, the pressure force is P˜δA . These re- k Thewaterbridgeconsistsofaflexiblefluidcablewhich sultsdescribecompletelywhentousethepressureP and when to use the pressure P˜. can be suspended by its endpoints. There is a slight sag inthecableasbefitsthe equilibriumofthetotalgravita- tionalforceMg downwardandthetotalMaxwelltension force 2τsinθ upward wherein θ is the angle between the IV. TENSION IN A FLUID CYLINDER cable tangent at the support and the horizontal; For a fluid dielectric cylinder of length L and cross 2πρgL Mg =2τsinθ ⇒ sinθ = (28) sectional area A in a uniform electric field parallel to (ε−1)E2 the axis, let us consider the work done in changing the volume V =LA via wherein Eq.(3) has been invoked. Eq.(28) allows for the theoretical computation of θ in terms of experimental dV =AdL+LdA. (23) dielectric constants, mass densities, water bridge lengths and electric fields. With regard to the electric field, one Employing the pressure tensor Eq.(22), one finds that notes the exact conversion between the Gaussian units both pressures, P˜ and P are required here employed and the usual engineering units, dW =−PÃdL−PLdA. (24) 1 Gauss≡299.792458volt/cm. (29) WhenafluidisstretchedatconstantvolumedV =AdL+ The agreement between theory and experiment in pre- LdA=0 dicting the slight hang of the water bridge as a flexible −LdA=AdL ⇒ dW =(P −P˜)AdL=τ˜dL. (25) cable, as in the above Fig.1, is satisfactory. The effective tension is thereby VI. HYDROSTATICS IN AN ELECTRIC FIELD εE2A τ˜=(P −P˜)A= , (26) 4π Theforcedensityonthefluidasdescribedbythepres- sure tensor Eq.(22) is given by wherein Eq.(12) has been invoked. Subtracting the tension that would be present for an electric field in the εEE vacuum, τ =τ˜−τ , yields our final result for the ten- f =−divP=−gradP +div . (30) vac (cid:18) 4π (cid:19) sion, Within the bulk liquid divD = div(εE) = 0 so that 2 (ε−1)E A τ = (27) Eq.(30) thereby reads 4π ε in agreement with Eq.(2) of Sec.I. f =−gradP + (E·grad)E. (31) 4π 4 Onthe otherhand, fromthe Gibbs-DuhemEq.(8) under VII. BERNOULLI FLOWS IN STRONG FIELDS equilibrium isothermal conditions dT =0, ε We here employ the usual notion of a fluid derivative gradP =ρgradζ+ (E·grad)E (32) 4π operator, so that the force per unit volume can be computed from d ∂ the chemical potential per unit mass; i.e. = +(v·grad), (39) dt ∂t f =ρgradζ. (33) which expressesthe time rate ofchange operatoras seen Under a Newtonian gravitationalfield by an observer moving locally with the fluid velocity v. For example, mass conservation reads[6] g=−gradΦ, (34) dρ one finds the total force density equilibrium condition =−ρ divv. (40) dt f +ρg=−grad(ζ+Φ)=0 (35) Bernoulli flows are adiabatic, i.e. viscous entropy pro- yielding the uniform chemical potential condition ductionisignored. Conservationofenergythenamounts to a local entropy conservation law[6] 2 µ=m[ζ(ρ,T,E )+Φ]=const. (36) ds To compute the electric field dependence of the chemi- =−s divv. (41) dt cal potential per unit mass, one may apply a Maxwell relation to the thermodynamic Eq.(7) which reads whereins is the entropyper unit volume. FromEqs.(39) and (40) it follows that ∂ζ 1 ∂D α =− =− TE, (37) (cid:18)∂E(cid:19) 4π (cid:18)∂ρ (cid:19) m d s 1 ds dρ T,ρ T,E = ρ −s =0, (42) dt(cid:18)ρ(cid:19) ρ2 (cid:18) dt dt(cid:19) wherein the mean molecular dipole moment p = α E T defines the polarizability αT as in Eq.(4). Integrating whichimpliesaconservationlawfortheentropyperunit Eq.(37) completes the calculationof the chemicalpoten- mass tial s ds∗ 1 E2 2 2 s∗ ≡ ρ ⇒ dt =0. (43) ζ(ρ,T,E)=ζ0(ρ,T)− αT((ρ,T,F ))d(F ), 2mZ0 It is here that the enthalpy per unit mass w makes it’s α (ρ,T,E2 =0) ζ(ρ,T,E)=ζ0(ρ,T)− T E2+... (.38) way into the adiabatic polar dielectric liquid Bernoulli 2m flows in an electric field; ThecentralEq.(38)ofthissectionimpliesthatthechem- ζ =w−Ts∗, ical potential is lowered when strong electric fields are 1 1 applied. dw =Tds∗+ dP − D·dE, (44) It followsfromEq.(38) that the applicationofanelec- ρ 4πρ tric field lowers the chemical potential of films of water wherein Eq.(8) has been invoked. For an adiabatic flow adsorbed on insulating substrates such as glass which is with ds∗ =0 as in Eq.(43), Eq.(44) implies oftenemployedinphysicalchemistryexperiments. When the chemical potential of a liquid film is lowered, the 1 ρgradw =gradP − (D·grad)E. (45) water film thickness increases; e.g. in the presence of 4π electric fields, water in a glass beaker will have a film Thus,the Maxwellpressuretensorforce per unit volume which appears to climb higher up the walls than would Eq.(31) in an adiabatic Bernoulli flow reads bepossibleinthezeroelectricfieldcase. Thefabrication of a water bridge begins by applying a potential differ- f =−ρ gradw. (46) ence acrossthe water contained in two neighboring glass beakers which just touch each other. One expects and The dynamical equation of motions which accounts for finds experimentally a thickening film layer all around momentum conservation then becomes the points at which the water horizontal surfaces meet dv the two beaker walls. The water climbs the walls of ρ =f +ρg, dt both beakers and near the touch point of the beakers dv splashes over the tops. When the hydrostatic calm after =−grad(w+Φ), (47) thesplashbegins,abridgeisformedatthepointwherein dt the beakers just touch. A longer water bridge is formed wherein the gravitational field Eq.(34) has been taken after slowly separating the beakers. Let us now turn to into account. Vorticity the hydrodynamic features of the polar liquid flows in a strong electric field. Ω=curlv (48) 5 makes an appearance in virtue of the acceleration iden- For an adiabatic (ds∗ = 0) incompressible (dρ = 0) tities steadystateBernoulliflow,Eqs.(53),(54)and(55)imply the central result of this section dv ∂v = +(v·grad)v, dt ∂t 2 1 ǫE dv = ∂v +Ω×v− 1grad(v2), (49) P + 2ρv2+ρgz− 8π =const. (56) dt ∂t 2 which allow us to write Eq.(47) as In the case that E = 0, the Bernoulli Eq.(40) indicates thatwaterundertheinfluenceofgravityaloneflowsdown ∂v 1 2 awallandmaysplashatthebottom. Ontheotherhand, +Ω×v=−grad w+Φ+ v . (50) ∂t (cid:18) 2 (cid:19) ifthe electricfieldonthe bottomofthe wallis negligible and the electric field on top of wallis large,then a polar Employing the curl of Eq.(50) and using Eq.(48) implies dielectricfluidcancrawlupawallandmaysplashatthe the equation of motion for vorticity. It is top. Such top splash processes on both of two beakers has been a precursor for building a water bridge. ∂Ω +curl(Ω×v)=0, ∂t dΩ =(Ω·grad)v−Ω(divv). (51) dt VIII. CONCLUSIONS IfatagiveninitialtimetheBernoulliflowisirrotational, i.e. Ω = 0, then at all later times in accordance with Water subject to high electric fields can sustain struc- Eq.(51) the flow will remain irrotational. Eq.(50) then tures that are more than just a bit unusual. An electric reads[6] fielddirectedparalleltothewatercylinderaxiscancreate a tension as in a stretched rubber band but with differ- v=gradϕ, ent causes. In the case of the rubber band, the tension ∂ϕ + 1|gradϕ|2+w+Φ=0, arisesfrom the high entropyof random knotted polymer ∂t 2 chains. In a polar liquid the tension arises out of long ∂v 1 ordered chains of low entropy aligned coherent dipolar =−grad w+Φ+ v2 . (52) ∂t (cid:18) 2 (cid:19) domains [7, 8, 9, 10, 11]. The resulting tension in the waterbridgesustainsasiphontube betweentwobeakers ThecompletesetofequationsforasteadystateBernoulli withouttherequirementthatanewexternalsiphontube flow in a strong electrostatic field E and in a uniform structure of other materials be introduced. gravitational field g in the negative z-direction then fol- Weshallconcludewithsomecloseanalogiesbetweena lows from Eq.(52) as polar dielectric fluid, such as water, and a quantum superfluid suchas liquid 4He. In the liquid 4He superfluid 2 2 curlE=0, case, it is known that the superfluid film can act simi- divD=div(εE)=0, larly to a siphon in which the film climbs over the wall 1 and down the other side of the wall until the lowestpos- w(s∗,P,E)+ |v|2+gz =const. (53) sible gravitational energy is achieved. Similar flows can 2 be induced in water by the application of high electric The only difference between the normal steady state fields. In both cases, the flows are carried with little en- Bernoulli fluid flows in Eq.(53) and the usual case for tropyproduction,althoughinthecaseofdeionizedwater E = 0 resides in the electric field contributions to the there is a mild Ohmic heating due to the transport of a enthalpyperunitmassw. Itishereusefultointroducea few remaining ions. new thermodynamic potential per unit mass ̟ obeying Finally, when the vorticity Ω becomes important[12], as in the boundary layer between the wall and the P w =̟+ , (54) Bernoulli flow, one may expect quantum vortex lines in ρ waterwiththecirculationcondition[13] v·dr=2π¯h/m. d̟ =Tds∗+ P dρ− 1 D·dE, The reasoning is that in a coherentHliquid flow, the ρ2 4πρ velocity potential ϕ in Eq.(52) determines the phase ̟(s∗,ρ,E2)=̟0(s∗,ρ)− factor exp[i(m/¯h) jϕ(rj,t)] in the many body fluid 1 E2 wave functions. TPhis quite general connection between ε(s∗,ρ,F2)d(F2), the Bernoulli flow Eq.(52) and the quantum mechanical 8πρZ0 phase is not self evident[14]. For this reason we have ε(s∗,ρ,0)E2 placed the mathematical details of the proof in the at- ̟(s∗,ρ,E2)=̟0(s∗,ρ)− +... . (55) tached AppendixA. 8π 6 APPENDIX A: QUANTUM FLUID PHASE tion with the Bernoulli velocity potential ϕ(r,t) deter- mining the phase, one notes that Thevelocitypotentialϕ(r,t)intheBernoulliEqs.(52) and (55), ρˆ′ =U†ρÛ =ρ, Jˆ′ =U†JÛ =Jˆ+ρˆgradϕ, (A6) ∂ϕ 1 2 P +ρ + |gradϕ| +Φ+̟ =0, (A1) (cid:20)∂t 2 (cid:21) the Bernoullivelocityv=gradϕ contributesto the current leaving the density unchanged. may be employed in the phase of the many body fluid The unitary transformation on the Hamiltonian con- quantum wave functions, tains a time derivative term as given by U(t)=expi¯hm ϕ(rj,t), Hˆ =U†HÛ −i¯hU†∂∂Ut , Xj   Hˆ =Hˆ + Jˆ·gradϕd3r i 3 Z U(t)=exp ρˆ(r)ϕ(r,t)d r , (A2) (cid:20)¯hZ (cid:21) ∂ϕ 1 2 3 + ρˆ + |gradϕ| d r, (A7) wherein r is the position of the jth molecule and the Z (cid:20)∂t 2 (cid:21) j quantum mechanical field operator for the mass density leading finally to is given by[15] ρˆ(r)=m δ(r−r ). (A3) Hˆ =Hˆ −E + Jˆ·gradϕd3r. 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