Theory of the Maxwell Pressure Tensor and the Tension in a Water Bridge

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by  A. Widom. J. Swain

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 chem-
ical 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 =−P˜AdL−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 ten-
sion 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 su-
perfluid 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†ρˆU =ρ,
Jˆ′ =U†JˆU =Jˆ+ρˆgradϕ, (A6)
∂ϕ 1
2
P +ρ + |gradϕ| +Φ+̟ =0, (A1)
(cid:20)∂t 2 (cid:21) the Bernoullivelocityv=gradϕ contributesto the cur-
rent 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†HˆU −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. (A8)
j Z
Xj In the final Hamiltonian Eq.(A8), the Bernoulli Eq.(52)
has been invoked with a total enthalpy
We presumea fluid microscopicHamiltonianofthe form
Hˆ = 21m |pˆj|2+V in which pˆj =−i¯hgradj (A4) E =Z ρ(w+Φ)d3r. (A9)
Xj
The total enthalpy may be found in the microcanoni-
and the positions {rj} all commute with V. The field cal ensemble via the eigenvalue problem Hˆ |ni = En|ni.
operator for the mass current density is thereby[15] Diabatic entropy producing terms producing vorticity in
boundary layers from the Bernoulli irroationalflow arise
1
Jˆ(r)= {pˆ ,δ(r−r )} (A5) from the interaction Hamiltonian
j j
2
Xj
Hˆint = Jˆ·gradϕd3r. (A10)
wherein the curly brackets indicate an anti-commutator; Z
{a,b}≡ab+ba. When viewed as a unitary transforma-
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