On the failure of water to freeze from its surface

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On the failure of water to freeze from its surface

Michael Elbaum, M. Schick

To cite this version:

Michael Elbaum, M. Schick. On the failure of water to freeze from its surface. Journal de Physique I,

EDP Sciences, 1991, 1 (12), pp.1665-1668. �10.1051/jp1:1991233�. �jpa-00246443�

f

Phys.

I France 1

(1991)

1665-1668

DtCEMBREI991,

PAGE 1665

Classification

PhysicsAb5tracts

64.70D-68.42

Show Coululunication

On the failure of water to freeze from its surface

Michael Elbaum and M. Schick

Department

Of

Physics FM-15, University

of

Washington,

Seattle WA

98195,

U.S.A.

(Received12

August 1991,

uccepted

9

October1991)

Abstract. We show that _water in

equilibrium

is _not

expected

to freeze %om its surface

inward,

contrary to the

expectation

of

elementary

theories. We do this in _two ways;

by utilizing

a recent

calculation of the interfacial energy of the icefivater interface, and

by

an

explicit

calculation of the surface free energy of _a

water/vapor

interface which

incorporates

a film of ice.

It is a familiar observation that _{as a}

body

of water

freezes,

a film of ice appears ^{at its} surface.

As latent heat is

removed,

this film thickens until all the water is frozen. That the ice is found at the interface between water and

vapor

^is

due,

of course, to the fact that its

density

is intermediate

between them. An

interesting question

is

whether,

in the absence of

gravity,

ice would

appear

^at

the water

surface,

due to forces within the water itselL Here we consider the influence of such

forces on the

freezing

of water, and concentrate on the

question

^{of whether the surface itself}

nucleates the

freezing

transition.

The usual

apprqach

to such _a

question

is to determine whether

replacing

a

given

thickness of the

liquid

at the interface

by

an

equal

thickness of solid raises or lowers the

system's

free energy.

The

change

in the free energy ^{due to} this

replacement

is related to the difference in the

strengths

of the interaction between correlated

dipole

fluctuations ix the two

phases,

that

is,

to the difference in their _van der Waals forces. In

determining

this

difference,

various _common

approximations

are

often introduced. One is to assume that the difference in the interaction

strengths

is

proportional

to the difference in densities of the

phases (I,e,

^{that the} bulk

polarizability

is

simply

a sum of molecular

units);

another is to characterize it

by

the difference in the visible refractive indices.

Either

assumption

leads to the conclusion that the surface free energy ^{of the}

waterlice/vapor

_sys-

tem is _a

positive

and

monotonically decreasing

function of the thickness of the ice

layer.

It follows that an

arbitrarily

thick

layer

of ice should form at the water

vapor

^{interface at the}

freezing point.

Furthermore,

at

temperatures

^close to, ^{but above the}

freezing point,

a thin film of ice should

form,

whose thickness grows ^without limit _as the

freezing temperature

^is

approached

from above.

This

phenomenon

is known _as "surface

freezing [I]".

An immediate consequence ^{is that the} su-

percooling

^of water with _a free surface should be

extremely difficult,

because the interface itself would nucleate

freezing.

This

is,

of _course,

contrary

to common observation.

Hence,

the usual

approximations

^fail.

1666 JOURNAL DE PHYSIQUE I N°12

Here _we utilize the more

complete theory

of

dispersion forces,

due to

Dzyaloshinskii, Lifshitz,

and Pitaevskii

[2] ~I>LP)

to

approach

the

question

of the surface

freezing

ofwater. This formalism

includes the full

frequency dependence

of the van der Vbals forces in the

system,

a

dependence

which _we

recently

found

[3]

to be

extremely important

when

considering

the related

question

of the "surface

melting"

^of

ice,

I.e. whether ice melts inward from _a thin

liquid

film nucleated

by

surface forces at

temperatures

be&Yw the bulk

melting temperature.

^In that case, we found that ice does not surface melt

completely,

but is

quite

close to

doing

_so as shown

by

the fact that a rather thick

liquid layer

h

predicted

to

appear

on it at the

triple point.

The existence of this thick film could be traced to the

high frequency components

^{of the}

fluctuating

_van der Waals interaction.

This _nearness of ice to surface

melting

is also indicated

by

the relation between the

ice/Vapor,

icefivater,

and

water/vapor

interfacial

energies

~Iv = «iw + _«wv + b.

(1)

If ice did in fact surface

melt,

then b would be

identically

zero. We found it to be non-zero, but

extremely small,

b

= -4 _x

10~~ erg/cm2,

orders of

magnitude

smaller than all three of the interfacial

energies

which _are between

101

and

102 erg/cm2.

A small

negative

value of b is also

implied by experiments _[4] showing

_a

small,

but non-zero, contact

angle

of water

droplets

_on ice at the

triple temperature.

This result _can be used to rule out the surface

freezing

of water as follows. If this process ^did take

place,

it would

imply

that the interfacial

energies

_were related

according

to

awv = aIw + _«Iv.

(2)

Using equation (I),

one finds that this would

imply

b ₌

-2aIw,

which from _our calculated b is

clearly

not true.

Hence,

the surface

freezing

ofwateris excluded

[5].

^{We illuminate this conclusion} below

by

a direct

application

^{of the}

theory

of

dispersion

^forces to this

system,

^{and show} that when the full

frequency dependence

of the

electromagnetic _response

is

included,

the van der Waals

interactions lead _{one to}

expect

no ice at all _to form at the _water

vapor

^interface.

We consider _a

system

^of water and

vapor

ⁱⁿ coexistence at the

triple temperature

^with a film of ice of thickness L at the interface. We calculate the excess Helmholtz free energy per ^unit area, and determine the

equilibrium

thickness of the film of ice as that value which minimizes this excess

free energy. This _excess

quantity

can be written _{as aIw + alv +}

F(L)

where the contribution of the

dispersion

forces to

F(L)

is

given by _[2]

~~~~ ~8~2 /j~

^{~~ ~~~~~~}

^~

+

~~~~

+

~~

~ ~~

+

lniI )ill I till)(ill I _till) ^e-~ii, (3J

with

~M =

[~2 r((1 _{~~')]~/~, (M}

=

I, V).

fw

In this

expression,

the dielectric functions _< of water

(W),

ice

(I),

^and _vapor

(V),

are evaluated at the

sequence

^of

imaginaTy frequencies I(n

₌

I(2~kT/h)n,

the

prime

on the _{sum means} that the

term _{n =} 0 receives _a

weight of1/2,

and _rn

=

2Lfn (<w)~/~/c,

with

k, h,

and _c the usual funda- mental constants. We take the dielectric function of vapor to be

unity.

The dielectric functions

N°12 ON THE FAILURE OF WATER TO FREEZE FROM ITS SURFACE 1667

required

in the above are obtained

by fitting

measurements of the

complex

dielectric functions

<(w)

^of ice and water to a

damped

oscillator forrn

~~~~ ^~

~

ej

hw~~

(hw)2'

^~~~

i

water

- 1.6

~ce

4li

#

1-z

1

0 1 Z

iOgl0 htn/~v

Fig-

I. Fits of the dielectric fi~nctions ofwater and of

ice,

evaluated at the discrete

imaginary %equencies I(n

used in

equation (3).

o

"

~

02

O

'

b0 04

~i

~~~~

~ 06

.0002

~-4

- ¹ _o

L nm]

Fig-

Z The contribution

F(L

to the

water/vapar

interracial energy as a fi~nction of the thickness L of _an

incorporated

ice film.

where the

ej, fj

and gj ^are

fitting parameters.

^{The static dielectric} constants are treated

sep- arately.

The _sources of the

experimental

data _as well _{as a} table of the

fitting parameters

^which

result _are

given

in reference

[2],

whfle the

resulting

dielectric functions of ice and _{water are} shown in

figure

I.

Insening

these functions into

equation (I),

_we obtain for

F(L),

the contribution of

the _van der llbals interactions to the _excess Helmholtz free energy per unit area, the result shown

1668 JOURNAL DE PHYSIQiJE I N°12

in

figure

2. The

global

minimum of this function

clearly

occurs at L

= 0

corresponding

to no film of ice at all at the water vapor

interface,

and to _an

infinitely negative

value for

F(L). However,

the minimum value of

F(L) is, by definition, equal

to awv aIw alv, which is bounded from below

by -2aIw.

This follows %om

equation (I)

and the fact that b cannot be

positive.

If _we ask at what thickness

F(L)

attains this lower

bound,

which b _on the order of -50 erg

/cm~,

_we find _a thickness of 0.03 nm

again corresponding

to no film of ice. The continuum

theory employed

here is not

applicable

at such

distances,

_so that _a molecular thickness of ice _can not be ruled out, but this does not alter _our result that the _van der Waals forces alone do not favor the existence of _a thick film of ice at the water

vapor

^interface.

Again,

this result _can be traced to the

high frequency components

^{of the dielectric functions shown in}

figure I;

in

particular

to the fact that the dielectric

response

^of

ice,

evaluated at

imaginary frequencies greater

^than

iwo

with _{wo *} 2 _x 101~

rad/s,

is

greater

^{than that of water in the} same range. Had _we used

only

low

frequency data,

_say visible and

below,

we would have obtained _a

positive, monotonically decreasing F(L)

^with its minimum at infinite

L,

as in the

simplified theory

^referred to

earlier,

which would have indicated surface

freezing.

Note that there is

actually

a minimum in the calculated surface free energy ^{at infinite}

thickness, (the

weak maximum is at L _m 5 nm, see

Fig. 2),

but use of the full

frequency

_response

shows it _to be _a

local,

rather than _a

global,

minimum. We _can have confidence in this result be-

cause our calculated

F(L)

crosses zero at

3.2

_nm,

sufficiently

thick _on the atomic scale to

satisfy

the

assumptions

^{of the} calculation~

In sum, we have shown that

contrary

to the

expectation

of

elementary approximations,

water in

equih~rium

should not

undergo

surface

freezing;

that

is,

above the

freezing temperature,

^{ice is}

not

expected

to be nucleated at the water surface and to thicken

progressively

as the

temperature

is lowered. TMs is in accord with the

commonly

observed

supercooling

of water.

Acknowledgeulents.

We thank J.G. Dash for many

stimulating

conversations. Thb work was

supponed

in

part by

the National Science Foundation under Grants No.

DMR-891tiJ52,

No.

DMR-8913454,

and No.

DPP-9023845.

llefemnces

ii]

For _a review of Ihe

analogous

process ^{of Surface}

melting,

See DIErRICH

s.,

in Phase Transitions and Cdtical

Phenomena,

C. Domb and J. Lebowitz Eds. vol 12

(Academic,

New York~

19B8)

p.

137,

or DASH J.G, in

Proceedings

of the Nineteenth

solvay Conference,

EW de l&btte Ed.

(springer- Verlag,

Berlin,

1988).

[2] DzYMmHINSKII

I-E-,

LIPSHnz E.M, and PrrABVSKII

L-P,

Adv

Phys.

lo

(I%1)

^165.

[3] ELBAUM M. and scHIcK

M.,

_P%ys. Rev Le1L 66

(1991)

1713.

[4] Kmc- WM. and HOBBS

PV,

Ph&s.

Map

19

(1%9) l161;

KNIGHT

CA.,

Phiks.

Map

23

(1971)

^153.

[5l^We are indebted to Michael Wortis for

reminding

_us of these

triangle inequalities.