Frequency-Dependent

Properties

of

a

Fluid

Jet

Stimulus

Calibration

Modeling

and

Application

to

Cochlear

Hair

Cell

Bundles

The

investigation

of

small

physiological

mechano-sensory

systems

such

as

hair

cells

or

their

accessory

structures

in

the

inner

ear

or

lateral

line

organ

requires

mechanical

stimulus

equipment

that

allows

spatial

manipulation

with

micrometer

precision

and

stimulation

with

amplitudes

down

to

the

nanometer

scale

Here

we

describe

the

calibration

of

a

microfluid

jet

produced

by

a

device

that

was

designed

to

excite

individual

cochlear

hair

cell

bundles

or

cupulae

of

the

fish

superficial

lateral

line

system

The

calibration

involves

a

precise

definition

of

the

linearity

and

time

and

frequency-dependent

characteristics

of

the

fluid

jet

as

produced

by

a

pressurized

fluid-filled

container

combined

with

a

glass

pipette

having

a

microscopically

sized

tip

acting

as

an

orifice

A

procedure

is

described

that

can

be

applied

during

experiments

to

obtain

a

fluid

jet

's

frequency

response

which

may

vary

with

each

individual

glass

pipette

At

small

orifice

diameters

-LRB-

μm

-RRB-

the

fluid

velocity

of

the

jet

is

proportional

to

the

displacement

of

the

piezoelectric

actuator

pressurizing

the

container

's

volume

and

is

suitable

to

stimulate

the

hair

bundles

of

sensory

hair

cells

With

increasing

diameter

the

fluid

jet

velocity

becomes

proportional

to

the

actuator

's

velocity

The

experimentally

observed

characteristics

can

be

described

adequately

by

a

dynamical

model

of

damped

fluid

masses

coupled

by

elastic

components

INTRODUCTION

Sensory

hair

cells

are

the

primary

mechano-detectors

in

several

mechano-sensory

organs

in

vertebrates

such

as

the

hearing

and

vestibular

organ

and

the

lateral

line

system

in

fish

and

amphibians

The

hair

bundle

is

the

mechano-sensitive

organelle

of

a

hair

cell

and

consists

of

a

well-organized

arrangement

of

villi

on

the

apical

side

of

the

cell

called

stereocilia

in

some

types

of

hair

cells

together

with

a

true

cilium

the

kinocilium

In

most

mechano-receptor

organs

the

hair

cells

are

covered

by

an

overlying

tectorial

structure

bathed

in

fluid

often

mechanically

coupling

many

hair

bundles

-LRB-

Pumphrey

van

Netten

-RRB-

These

tectorial

structures

when

driven

by

fluid

flow

convey

the

mechanical

signals

present

in

the

fluid

flow

to

the

hair

bundles

In

the

mammalian

cochlea

for

instance

sound

signals

are

converted

to

vibrations

of

the

cochlear

endo

and

perilymph

which

drive

the

cochlear

partition

causing

a

motion

of

the

stereocilia

Deflections

of

a

bundle

of

stereocilia

which

pivot

at

their

base

induce

changes

in

the

open

probability

of

mechanically

gated

ion

channels

located

in

the

tips

of

the

stereocilia

-LRB-

e.g.

Hudspeth

Fettiplace

and

Ricci

-RRB-

The

combined

mechanics

and

hydrodynamics

of

the

hair

bundles

and

tectorial

structure

determine

an

organ

's

sensitivity

to

a

specific

aspect

of

the

excitatory

fluid

motion

-LRB-

e.g.

van

Netten

-RRB-

To

quantitatively

study

the

mechanics

of

tectorial

structures

and

individual

hair

bundles

and

the

mechano-electrical

transduction

process

of

hair

cells

a

well-defined

mechanical

stimulus

is

required

Due

to

the

small

dimensions

of

most

tectorial

structures

manipulation

at

a

microscopic

spatial

resolution

is

desirable

especially

when

stimulating

an

individual

hair

bundle

A

commonly

used

mechanical

stimulus

for

individual

hair

bundles

consists

of

an

electromagnetic

or

piezoelectric

actuator

driving

an

elongated

microscopic

fiber

or

glass

probe

pushing

against

the

hair

bundle

-LRB-

e.g.

Strelioff

and

Flock

Crawford

and

Fettiplace

Howard

and

Ashmore

Howard

and

Hudspeth

Russell

et

al.

Kennedy

et

al.

-RRB-

The

probe

's

tip

may

adhere

to

the

cell

membrane

of

the

kinocilium

or

tallest

stereociliar

row

allowing

for

force

application

to

the

bundle

in

both

directions

A

stiff

probe

enables

direct

displacement

of

the

bundle

with

virtually

the

same

amplitude

as

the

driving

element

overruling

the

intrinsic

bundle

mechanics

and

can

be

used

to

study

transducer

channel

kinetics

-LRB-

e.g.

Holton

and

Hudspeth

Assad

and

Corey

Vollrath

and

Eatock

Kennedy

et

al.

-RRB-

Alternatively

a

more

compliant

probe

with

a

stiffness

comparable

to

that

of

the

hair

bundle

can

be

used

Although

under

such

conditions

the

application

of

displacement

may

become

less

direct

the

stiffness

of

the

probe

can

be

designed

as

to

not

dominate

the

intrinsic

mechanics

of

the

hair

bundle

Consequently

it

allows

for

estimating

the

bundle

's

intrinsic

mechanical

characteristics

like

its

compliance

or

resistance

-LRB-

Crawford

and

Fettiplace

Howard

and

Ashmore

Howard

and

Hudspeth

Jaramillo

and

Hudspeth

Benser

et

al.

Kennedy

et

al.

-RRB-

Viscous

forces

do

however

attenuate

the

efficiency

of

a

compliant

probe

at

higher

frequency

stimuli

limiting

their

use

up

to

a

certain

cut-off

frequency

-LRB-

Crawford

and

Fettiplace

Howard

and

Hudspeth

-RRB-

As

several

types

of

hair

cells

and

most

mechano-detecting

structures

in

mechano-receptor

organs

are

fluid-driven

a

fluid

coupling

between

stimulator

and

mechano-receptor

is

an

attractive

natural

alternative

to

direct-contact

stimulation

with

a

probe

Well-defined

fluid

displacements

can

be

obtained

using

a

stimulus

sphere

attached

to

a

piezoelectric

element

Submerged

in

water

a

sphere

vibrating

with

a

constant

displacement

amplitude

may

generate

a

near-field

water

flow

that

has

a

constant

displacement

amplitude

as

a

function

of

frequency

up

to

the

order

of

a

kilohertz

-LRB-

van

Netten

-RRB-

It

does

however

stimulate

very

inefficiently

at

larger

distances

as

the

stimulus

strength

in

the

near

field

measured

as

water

displacement

amplitude

-LRB-

A

-RRB-

is

proportional

to

the

third

power

of

the

ratio

of

stimulus

sphere

radius

-LRB-

a

-RRB-

to

the

distance

-LRB-

r

-RRB-

to

its

centre

-LRB-

i.e.

e.g.

van

Netten

-RRB-

To

compensate

for

this

reduced

efficiency

at

larger

distances

a

stimulus

sphere

with

larger

radius

-LRB-

a

-RRB-

may

be

used

Large

spheres

however

are

less

practical

in

many

situations

in

which

small

experimental

chambers

and

objective

lenses

do

not

provide

sufficient

space

To

obtain

a

more

targeted

but

sufficiently

strong

mechanical

stimulus

a

fluid

jet

emerging

from

a

glass

pipette

with

a

small

opening

at

its

tip

may

provide

a

practical

alternative

solution

The

use

of

a

fluid

jet

to

mechanically

excite

individual

hair

bundles

may

have

several

advantages

over

stimulating

with

glass

probes

noticeably

yielding

a

better

defined

resting

position

of

the

bundle

a

more

even

application

of

force

and

in

the

case

of

lymph-driven

hair

cells

a

more

natural

physiological

stimulus

Also

the

asymmetry

in

force

that

may

occur

in

pushing

and

pulling

the

cell

membrane

of

the

stereocilia

inevitably

related

to

the

direct

contact

of

a

glass

probe

can

be

avoided

It

is

feasible

that

glass

probes

allow

wider

stimulating

bandwidths

although

-LRB-

lateral

-RRB-

resonances

have

been

reported

that

may

limit

the

high

frequency

response

to

the

order

of

kHz

-LRB-

e.g.

Beurg

et

al.

see

also

Discussion

section

-RRB-

Fluid

jet

stimuli

have

been

considered

as

effective

force

stimuli

whereas

-LRB-

stiff

-RRB-

probe

excitation

present

displacement

stimuli

to

the

bundle

-LRB-

Vollrath

and

Eatock

-RRB-

Such

differences

in

stimulation

may

also

lead

to

differences

in

speed

of

engagement

of

the

transduction

apparatus

In

one

of

the

first

reports

on

the

use

of

a

fluid

jet

in

mechano-reception

research

it

was

applied

to

drive

the

cupula

of

a

fish

lateral

line

neuromast

-LRB-

Jielof

et

al.

-RRB-

The

pipette

opening

diameter

used

in

that

study

was

relatively

wide

compared

to

later

more

miniaturized

models

which

were

designed

to

displace

individual

hair

bundles

-LRB-

Flock

and

Orman

-RRB-

A

first

step

to

calibrating

the

frequency

dependence

of

a

fluid

jet

was

set

by

Saunders

and

Szymko

-LRB-

-RRB-

They

used

glass

micro-beads

with

a

density

close

to

that

of

water

that

were

captured

in

the

hydrodynamic

flow

field

and

monitored

their

motion

by

stroboscopic-illuminating

microscopy

Later

developments

of

fluid

jet-producing

devices

were

utilized

to

displace

single

inner

or

outer

hair

cell

bundles

or

lateral

line

cupulae

and

were

in

some

cases

pressure-calibrated

-LRB-

Kros

et

al

Denk

and

Webb

Géléoc

et

al.

Nicolson

et

al.

Vollrath

and

Eatock

-RRB-

In

the

present

work

a

calibration

procedure

of

a

newly

designed

piezoelectrically

driven

fluid

jet

device

is

presented

A

compliant

glass

fiber

with

a

resin

sphere

attached

to

its

tip

is

used

as

a

sense

probe

to

measure

the

dynamic

fluid

displacement

output

of

the

fluid

jet

device

The

motion

of

the

sense

probe

which

is

reflecting

the

dynamics

of

the

fluid

jet

is

monitored

using

a

displacement

measurement

method

implemented

in

a

transmitted-light

microscope

The

procedure

which

can

be

applied

in

an

experimental

setting

enables

the

determination

of

the

frequency

dependence

of

the

fluid

jet

This

is

essential

to

correct

experimental

data

obtained

using

the

same

fluid

jet

as

a

stimulus

To

understand

the

physical

factors

of

the

fluid

jet-producing

device

governing

the

frequency

response

of

the

produced

fluid

jet

a

theoretical

model

of

the

device

has

been

developed

The

results

of

the

model

were

compared

to

the

measured

data

and

show

that

the

fluid

resistance

of

the

orifice

at

the

pipette

's

tip

is

a

key

parameter

influencing

the

amplitude

and

phase

characteristics

of

the

emerging

fluid

jet

As

an

example

of

its

usefulness

the

fluid

jet

system

was

applied

to

induce

transducer

currents

in

mouse

cochlear

hair

cells

showing

the

importance

to

correct

for

the

phase

delay

of

the

fluid

jet

especially

at

high

frequencies

This

example

also

shows

that

fluid

jet

dynamics

may

lead

to

resonance

which

in

practice

limits

the

use

of

the

present

design

to

a

bandwidth

of

approximately

kHz

MATERIALS

AND

METHODS

Fluid

jet-producing

device

The

device

used

for

the

generation

of

a

fluid

jet

is

shown

in

Figure

and

its

individual

parts

in

Figure

The

main

parts

are

a

Perspex

fluid

container

-LRB-

C

-RRB-

and

a

brass

rear

end

-LRB-

R

-RRB-

Three

screws

-LRB-

S

-RRB-

pull

both

parts

together

sandwiching

a

piezoelectrically

driven

brass

disc

-LRB-

P

Ø

mm

thickness

mm

resonance

frequency

kHz

Conrad

Electronic

SE

Hirschau

Germany

-RRB-

in

a

configuration

that

is

clamping

the

disc

at

its

rim

The

Perspex

container

is

cone-shaped

and

is

closed

by

the

piezoelectric

disc

at

the

rear

end

A

rubber

o-ring

-LRB-

O

-RRB-

between

the

Perspex

and

the

piezoelectric

disc

provides

a

fluid-tight

seal

When

driven

by

a

voltage

signal

the

piezoelectric

disc

produces

forces

that

move

the

fluid

in

or

out

of

the

container

FIG.

The

fluid

jet-producing

device

A

Picture

of

the

assembled

fluid

jet-producing

device

mounted

on

an

x,y,z-micromanipulator

B

Disassembled

device

showing

its

separate

parts

R

Rear

brass

element

P

piezoelectric

disc

O

o-ring

S

screws

C

Perspex

container

W

rubber

cone

washer

Sp

Perspex

spacer

Sc

Perspex

screw

cap

Pi

glass

pipette

The

calibration

bar

indicates

cm

At

the

front

end

of

the

Perspex

container

a

glass

pipette

-LRB-

Pi

outer

Ø

mm

inner

Ø

mm

-RRB-

is

inserted

The

screw

cap

-LRB-

Sc

-RRB-

applies

pressure

on

two

cone

washers

-LRB-

W

-RRB-

separated

by

a

Perspex

spacer

-LRB-

Sp

-RRB-

to

produce

a

fluid

tight

seal

and

fixes

the

glass

pipette

in

a

mechanically

stable

way

In

most

applications

the

pipette

was

narrowed

at

its

tip

using

an

electrode

puller

Tip

diameters

were

varied

between

and

μm

and

tapering

lengths

were

kept

at

approximately

mm

Also

in

some

experiments

untreated

glass

pipettes

without

tapered

tips

were

used

so

that

their

effective

opening

equaled

the

inner

diameter

of

the

glass

pipette

-LRB-

Ø

mm

-RRB-

The

total

length

of

the

glass

pipette

was

about

cm

of

which

about

cm

was

clamped

inside

the

Perspex

container

To

facilitate

horizontal

application

of

the

produced

fluid

jet

in

the

experimental

chamber

the

glass

was

bent

cm

from

the

tip

over

an

angle

of

about

so

that

its

tip

could

be

aligned

within

the

horizontal

plane

-LRB-

Fig.

-RRB-

For

inducing

transducer

currents

in

hair

cells

-LRB-

see

Applications

-RRB-

straight

glass

pipettes

were

used

To

obtain

an

air-free

filling

of

the

fluid

container

in

the

Perspex

body

part

it

was

filled

with

a

low

viscosity

silicone

fluid

-LRB-

Fluid

CS

Dow

Corning

Midland

MI

USA

-RRB-

which

has

low

surface

tension

properties

The

silicone

fluid

was

degassed

under

vacuum

conditions

for

several

hours

before

use

to

prevent

dissolved

gas

from

creating

bubbles

inside

the

fluid

chamber

The

glass

pipette

was

filled

with

de-mineralized

water

or

in

the

case

of

physiological

experiments

with

extracellular

fluid

-LRB-

bath

solution

-RRB-

Inserting

the

filled

glass

pipette

into

the

Perspex

body

part

generated

a

pressure

increase

inside

the

fluid

container

replacing

a

volume

of

the

water

in

the

back

side

of

the

pipette

by

silicone

fluid

During

this

temporary

overpressure

the

fluid

jet

device

was

transferred

to

its

position

in

the

set-up

and

the

pipette

tip

was

submerged

in

the

bath

solution

preventing

air

from

entering

the

tip

When

prepared

as

described

above

the

instrument

also

benefits

from

the

high

resistivity

-LRB-

Ohm

cm

-RRB-

of

the

silicone

fluid

creating

an

electrical

resistance

between

the

brass

disc

and

the

bath

solution

thereby

preventing

electrical

cross-talk

of

the

piezoelectric

control

signal

to

the

possible

electrophysiological

recordings

obtained

from

a

preparation

Additional

electrical

shielding

was

obtained

by

connecting

the

brass

disc

which

is

facing

the

preparation

to

the

electrical

ground

The

rear

end

of

the

fluid

jet

device

which

together

with

the

grounded

brass

disc

enclosed

the

piezoelectric

material

was

also

grounded

thereby

shielding

off

the

applied

voltages

A

shaft

connected

to

the

rear

end

of

the

fluid

jet

device

was

used

to

mount

the

device

on

an

x,y,z-micromanipulator

-LRB-

Fig.

-RRB-

The

tip

of

the

jet

pipette

was

lowered

in

the

bath

solution

and

positioned

under

visual

or

microscope

guidance

All

calibration

measurements

were

done

approximately

mm

above

the

bottom

and

below

the

surface

Signal

generation

Sine

wave

stimuli

at

exponentially

distributed

frequencies

ranging

from

to

Hz

were

generated

at

points

per

period

using

the

full

amplitude

range

of

a

D/A

converter

-LRB-

Ariel

DSP

Highland

Park

NJ

USA

-RRB-

These

signals

were

subsequently

attenuated

to

the

desired

amplitude

and

filtered

-LRB-

pole

Bessel

Krohn-Hite

Corporation

Brockton

MA

USA

-RRB-

at

eight

times

the

stimulus

frequency

Frequencies

used

to

probe

resonances

exceeding

kHz

were

derived

from

a

signal

generator

-LRB-

PM

Philips

Eindhoven

The

Netherlands

-RRB-

Signals

for

exciting

hair

cells

were

generated

using

a

D/A

converter

with

dedicated

software

-LRB-

Power

and

Signal

software

package

Cambridge

Electronic

Design

Ltd

Cambridge

UK

-RRB-

Sense

probes

Sense

probes

used

for

the

detection

of

fluid

displacement

consisted

of

compliant

glass

fibers

-LRB-

Borosilicate

glass

capillaries

Harvard

apparatus

Ltd.

UK

-RRB-

produced

with

a

micropipette

puller

-LRB-

Sutter

instrument

Co.

Novato

CA

USA

-RRB-

with

a

small

sphere

that

was

produced

by

dipping

the

glass

fiber

tip

into

a

resin

-LRB-

Bison

Silicone

universal

transparent

Goes

The

Netherlands

-RRB-

After

curing

the

resin

sphere

diameters

ranged

from

to

μm

The

fibers

were

about

cm

in

length

with

a

gradually

decreasing

diameter

from

about

μm

to

approximately

μm

at

their

tips

The

fibers

were

glued

to

a

stiff

glass

capillary

that

was

subsequently

mounted

on

an

x,y,z-micromanipulator

The

mechanical

frequency

responses

of

the

sense

probes

to

fluid

displacement

were

obtained

by

stimulating

the

sense

probe

with

the

fluid

flow

produced

by

a

glass

stimulus

sphere

-LRB-

Ø

mm

-RRB-

attached

to

a

piezoelectric

element

The

frequency

response

of

this

stimulus

sphere

was

separately

determined

using

a

laser

interferometer

-LRB-

see

Motion

detection

-RRB-

When

stimulating

the

same

sense

probe

with

a

fluid

jet

the

center

of

the

fluid

jet

was

aimed

at

the

center

of

the

resin

sphere

The

distance

between

the

jet-producing

pipette

tip

and

the

resin

sphere

of

the

sense

probe

was

at

least

one

diameter

of

the

sphere

Based

on

the

frequency

responses

of

the

sense

probes

their

overall

stiffness

coupling

to

the

stiff

glass

tube

held

in

the

x,y,z-micromanipulator

was

estimated

to

be

of

the

order

of

N/m

Motion

detection

The

motion

of

the

stimulus

sphere

and

the

sense

probe

were

measured

with

a

heterodyne

laser

interferometer

coupled

to

a

fixed-stage

transmitted

light

microscope

mounted

on

a

vibration

isolated

table

The

objective

lens

-LRB-

WI

Zeiss

Oberkochen

Germany

-RRB-

of

the

microscope

-LRB-

modified

Axiotron

Zeiss

Oberkochen

Germany

-RRB-

focuses

two

coherent

laser

beams

-LRB-

HeNe

Newport

Spectra

Physics

GmbH

Darmstadt

Germany

-RRB-

aligned

in

parallel

so

that

their

beam

waists

intersect

in

the

plane

of

focus

creating

a

measuring

volume

with

microscopic

dimensions

-LRB-

Ø

μm

-RRB-

A

frequency

difference

of

kHz

between

the

two

laser

beams

produced

by

two

Bragg

cells

-LRB-

Isomet

Corporation

Springfield

VA

USA

-RRB-

driven

at

and

MHz

creates

a

moving

fringe

pattern

in

the

measuring

volume

Light

scattered

by

a

stationary

object

therefore

fluctuates

in

intensity

at

a

carrier

frequency

of

kHz

-LRB-

fhet

-RRB-

Additional

motion

of

the

object

induces

a

phase

modulation

of

these

intensity

fluctuations

which

is

displacement-dependent

The

back-scattered

laser

light

from

an

irregularity

on

the

surface

of

the

object

was

detected

by

a

photomultiplier

-LRB-

model

Hamamatsu

Photonics

Japan

-RRB-

coupled

to

an

I

V

converter

The

output

containing

the

carrier

frequency

fhet

phase-modulated

in

proportion

to

the

object

's

displacement

was

band-pass

filtered

with

a

centre

frequency

at

fhet

and

electronically

demodulated

using

a

modified

frequency

demodulator

-LRB-

OFV

Polytec

GmbH

Waldbronn

Germany

equivalent

r.m.s

velocity

noise

was

μm/s

-RRB-

Time-varying

phase

modulations

were

thus

treated

as

frequency

modulations

-LRB-

i.e.

Doppler

shifts

-RRB-

which

was

appropriate

as

the

phase

changes

per

time

of

the

carrier

were

small

compared

to

the

carrier

frequency

itself

Under

these

conditions

the

output

signal

of

the

demodulator

is

a

calibrated

linear

measure

of

the

velocity

of

the

object

within

the

range

of

to

μm/s

For

low

frequency

stimuli

-LRB-

Hz

-RRB-

a

digital

phase

demodulator

-LRB-

Polytec

GmbH

Waldbronn

Germany

-RRB-

was

used

which

has

an

output

proportional

to

the

displacement

of

the

object

with

a

resolution

The

demodulator

signals

were

low-pass

filtered

-LRB-

pole

Bessel

Krohn-Hite

Corporation

Brockton

MA

USA

-RRB-

at

eight

times

the

frequency

of

stimulation

and

were

amplified

before

being

digitized

using

a

A/D

converter

-LRB-

DSP

Ariel

Highland

Park

NJ

USA

-RRB-

with

a

sample

frequency

at

times

the

frequency

of

stimulation

Displacement

responses

consisting

of

usually

ten

consecutive

stretches

each

periods

in

length

were

averaged

on-line

by

the

data

acquisition

board

The

first

s

of

a

response

to

a

stimulus

were

not

recorded

to

prevent

effects

of

onset

transients

The

averaged

response

waveform

was

stored

on

hard

disc

A

fast-Fourier

transform

-LRB-

FFT

-RRB-

was

used

to

extract

the

amplitude

and

phase

at

the

frequency

of

stimulation

from

this

averaged

waveform

In

some

experiments

a

Position

Sensitive

Detector

-LRB-

PSD

On-Trak

Photonics

Inc

Lake

Forest

CA

USA

-RRB-

was

used

to

measure

displacements

of

uncalibrated

sense

probes

at

frequencies

beyond

kHz

using

a

A/D

converter

with

dedicated

software

-LRB-

Power

and

Signal

software

package

Cambridge

Electronic

Design

Ltd

Cambridge

UK

-RRB-

Hair

cell

electrophysiology

Experiments

were

performed

on

apical-coil

outer

hair

cells

obtained

from

acutely

isolated

organs

of

Corti

taken

from

neonatal

mice

-LRB-

Harlan

Netherlands

B.V.

Horst

-RRB-

Animal

procedures

conformed

to

Dutch

governmental

rules

and

the

guidelines

of

the

University

of

Groningen

Institutional

Animal

Committee

-LRB-

RuG-DEC

-RRB-

Transducer

currents

in

response

to

fluid

jet

stimulation

were

measured

under

whole-cell

voltage

clamp

with

an

Axopatch

patch

clamp

amplifier

-LRB-

Axon

Instruments

Inc

Molecular

Devices

Foster

City

CA

USA

-RRB-

at

mV

holding

potential

Further

details

on

the

procedures

and

solutions

used

can

be

found

elsewhere

-LRB-

Kros

et

al.

van

Netten

et

al.

-RRB-

Model

of

fluid

jet

dynamics

Schematics

of

a

mechanical

model

illustrating

the

principal

physical

elements

of

the

fluid

jet-producing

device

are

shown

in

Figure

The

device

is

shown

to

work

with

only

minor

nonlinear

distortion

-LRB-

e.g.

Fig.

-RRB-

so

that

the

system

may

be

adequately

analyzed

in

the

frequency

domain

In

this

section

all

dynamic

variables

will

therefore

be

considered

to

vary

harmonically

with

angular

frequency

ω

A

piezoelectrically

driven

disc

with

mass

Mpiezo

and

stiffness

Spiezo

is

used

to

pressurize

the

fluids

in

the

jet-producing

device

A

force

on

the

brass

diaphragm

of

the

piezoelectric

disc

is

produced

in

proportion

to

the

applied

voltage

This

causes

a

displacement

of

the

disc

Xc

so

that

the

fluid

in

the

Perspex

container

is

pressurized

Compressibility

of

the

fluid

is

neglected

-LRB-

see

also

Discussion

section

-RRB-

FIG.

Mechanical

model

representation

of

the

fluid

jet-producing

device

F

is

the

piezoelectrically

induced

force

proportional

to

the

applied

voltage

V

The

piezoelectric

element

consists

of

a

brass

disc

-LRB-

dark

gray

-RRB-

and

a

smaller

ceramic

plate

The

spring

S

piezo

and

a

resistive

element

R

c

together

with

the

mass

of

the

piezo

M

piezo

and

the

mass

of

the

container

fluid

M

c

-LRB-

cone-shaped

-RRB-

affect

the

fluid

displacement

amplitude

X

c

in

the

Perspex

container

The

fluid

displacement

X

c

is

amplified

through

a

lever

arm

with

ratio

L

into

L

X

c

which

drives

the

fluid

in

the

glass

pipette

The

same

lever

ratio

applies

to

the

forces

F

and

F

The

displacement

response

of

the

fluid

mass

in

the

pipette

X

p

is

determined

by

the

pipette

's

spring

S

p

the

pipette

's

tip

resistive

element

R

p

and

the

mass

of

the

pipette

fluid

M

p.

X

p

is

taken

as

a

proportional

measure

of

the

displacement

of

the

jet

fluid.FIG

Sense

probe

displacements

induced

by

the

fluid

jet

A

Sense

probe

displacement

in

response

to

a

vibrational

fluid

jet

stimulus

as

a

function

of

time

An

average

of

ten

traces

is

shown

B

Fast

Fourier

transform

of

the

average

waveform

shown

in

A

with

a

fundamental

component

at

Hz

and

a

second

harmonic

component

at

Hz

which

is

more

than

dB

less

in

amplitude

than

the

fundamental

component

Total

harmonic

distortion

-LRB-

THD

-RRB-

was

calculated

using

where

a

is

the

response

amplitude

at

the

fundamental

frequency

and

a

n

is

the

amplitude

of

the

nth

harmonic

component

as

determined

from

the

FFT

calculated

from

the

averaged

waveform

of

measured

sense

probe

responses

The

fluid

in

the

Perspex

container

has

a

mass

Mc

Fluid

mass

appears

in

the

dynamic

equations

-LRB-

e.g.

Eq

-RRB-

as

a

product

of

mass

times

its

acceleration

The

fluid

mass

in

the

cone-shaped

Perspex

container

can

be

considered

as

the

sum

of

a

series

of

cross-sectional

fluid

slices

with

variable

radius

so

that

the

fluid

mass

of

each

slice

is

proportional

to

the

square

of

its

radius

Assuming

that

the

equation

of

continuity

holds

in

combination

with

incompressibility

means

that

the

fluid

flux

along

the

series

of

slices

is

conserved

The

velocity

of

each

fluid

slice

and

also

its

acceleration

must

therefore

be

inversely

proportional

to

the

square

of

its

radius

so

that

the

product

of

mass

and

acceleration

of

each

slice

is

independent

of

radius

This

means

that

the

overall

product

of

mass

and

acceleration

of

the

fluid

in

the

cone-shaped

Perspex

container

can

be

dynamically

modeled

with

a

cylinder

with

length

-LRB-

mm

-RRB-

equal

to

the

cone

's

actual

length

and

a

diameter

-LRB-

mm

-RRB-

equal

to

the

cone

's

base

Therefore

the

mass

of

the

fluid

in

the

Perspex

container

is

modeled

as

and

its

effective

acceleration

as

Here

ρ

-LRB-

-RRB-

denotes

the

fluid

density

The

fluid

mass

in

the

container

Mc

-LRB-

kg

-RRB-

sums

with

that

of

the

piezoelectric

disc

Mpiezo

-LRB-

kg

-RRB-

resulting

in

Mtot

Mc

Mpiezo

-LRB-

kg

-RRB-

A

resistive

element

Rc

in

series

with

mass

Mc

is

included

which

is

related

to

friction

of

the

fluid

flow

in

the

Perspex

container

and

the

motion

of

the

piezoelectric

disc

The

mass

in

the

cylindrical

glass

pipette

is

described

by

-LRB-

kg

-RRB-

where

-LRB-

mm

-RRB-

and

-LRB-

mm

-RRB-

are

the

length

and

inner

diameter

of

the

glass

pipette

while

this

mass

has

a

displacement

Xp

The

overall

effect

of

the

conservation

of

flux

in

the

conical

Perspex

container

is

the

transformation

of

the

effective

displacement

of

fluid

in

the

Perspex

container

into

that

of

the

glass

pipette

The

transformation

factor

equals

the

ratio

of

the

-LRB-

maximum

-RRB-

cross-sectional

areas

of

the

two

fluid

compartments

-LRB-

-RRB-

The

force

ratio

of

this

transformation

is

governed

by

the

same

ratio

The

transformed

fluid

displacement

is

not

completely

transferred

into

displacement

of

the

fluid

mass

in

the

pipette

Xp

as

the

elastic

cone

washers

around

the

glass

pipette

absorb

it

partly

This

is

represented

by

the

compliant

element

Sp.

A

resistive

element

Rp

describes

the

viscous

friction

in

the

pipette

which

is

dominated

by

the

tip

resistance

The

resulting

fluid

displacement

Xp

is

taken

as

a

proportional

measure

of

the

fluid

displacement

of

the

jet

emerging

from

the

pipette

tip

Again

assuming

the

equation

of

continuity

this

proportion

is

related

to

the

square

of

the

ratio

of

the

pipette

's

tip

diameter

to

the

glass

pipette

's

cross-sectional

diameter

-LRB-

-RRB-

However

the

absolute

displacement

of

the

jet

is

strongly

dependent

on

the

distance

to

the

pipette

tip

-LRB-

See

also

Absolute

fluid

jet

velocity

in

the

Discussion

section

-RRB-

The

governing

equations

relating

the

piezoelectric

force

to

the

fluid

displacements

Xc

and

Xp

utilizing

the

usual

notation

for

time

derivatives

are

then

and

Changing

to

the

frequency

domain

Xc

and

Xp

are

all

assumed

to

vary

harmonically

with

time

so

that

they

are

proportional

to

the

factor

exp

-LRB-

iωt

-RRB-

This

factor

will

be

omitted

in

the

further

analysis

so

that

only

-LRB-

complex

-RRB-

amplitudes

can

be

written

for

these

variables

Displacement

of

the

fluid

emerging

from

the

jet

pipette

Xp

as

a

function

of

angular

frequency

ω

and

piezoelectrically

induced

force

is

then

described

by

with

a

relative

transformation

factor

T

-LRB-

ω

-RRB-

between

the

displacement

in

the

container

and

the

pipette

given

by

Inspection

of

the

denominator

of

Eq

shows

that

two

minima

may

occur

as

a

function

of

ω

corresponding

with

a

low

-LRB-

fl

-RRB-

and

high

-LRB-

fh

-RRB-

resonance

frequency

The

high

resonance

frequency

fh

can

be

found

by

minimizing

the

term

between

square

brackets

assuming

that

T

-LRB-

ω

-RRB-

varies

only

slowly

around

fh

This

frequency

is

therefore

related

to

Mtot

the

summed

mass

of

piezoelectric

disc

and

mass

of

the

fluid

in

the

Perspex

container

and

the

parallel

combination

of

Spiezo

and

the

up-transformed

stiffness

of

the

pipette

Together

they

amount

to

the

stiffness

in

accordance

with

Mtot

's

position

in

between

these

two

parallel

elastic

elements

-LRB-

Fig.

-RRB-

The

related

high-resonance

frequency

is

consequently

given

by

At

relatively

low

frequencies

the

frequency

dependent

terms

of

the

factor

between

square

brackets

-LRB-

Eq

-RRB-

are

relatively

small

so

that

this

factor

reduces

to

Spar

Then

Xp

can

be

approximated

in

the

low

frequency

range

by

where

Sser

is

the

series

spring

combination

of

Sp

and

Spiezo

given

by

It

then

follows

from

Eq

that

the

low-frequency

resonance

fl

is

given

by

Equation

shows

that

the

low

frequency

oscillation

is

due

to

the

mass

of

the

fluid

in

the

pipette

Mp

that

is

elastically

supported

by

the

effective

stiffness

Sser

which

is

formed

by

the

series

combination

of

Sp

and

the

down-transformed

piezo-stiffness

Spiezo

in

line

with

their

series

configuration

-LRB-

Fig.

-RRB-

The

parameters

Spiezo

Rc

Sp

and

Rp

were

determined

from

fitting

the

full

model

-LRB-

Eqs

and

-RRB-

to

the

measured

data

The

fluid

masses

were

determined

from

their

actual

physical

sizes

-LRB-

mm

mm

-RRB-

and

lengths

-LRB-

mm

mm

-RRB-

RESULTS

Linearity

of

the

fluid

jet

Figure

shows

a

typical

sinusoidal

displacement

of

the

sense

probe

driven

by

a

fluid

jet

produced

with

a

pipette

tip

diameter

of

μm

The

average

sense

probe

displacement

of

consecutive

traces

is

given

each

containing

periods

at

a

frequency

of

Hz

The

waveform

clearly

shows

the

periods

of

the

fundamental

frequency

with

negligible

harmonic

distortion

The

extent

of

the

distortion

was

determined

by

the

Fast

Fourier

Transform

-LRB-

FFT

-RRB-

calculated

of

the

waveform

shown

in

Figure

Figure

shows

the

resultant

frequency

content

of

this

waveform

up

to

Hz

The

fundamental

frequency

is

clearly

visible

at

Hz

and

rises

about

three

orders

of

magnitude

above

the

noise

floor

The

second

harmonic

component

is

visible

but

has

an

amplitude

more

than

dB

below

the

component

at

the

fundamental

frequency

Higher

harmonics

are

even

smaller

The

calculated

total

harmonic

distortion

in

this

case

was

dB

-LRB-

see

legend

Fig.

-RRB-

Fluid

jet

calibration

To

determine

the

frequency-dependent

motion

of

a

fluid

jet

produced

with

a

pipette

we

used

the

following

procedure

The

vibrational

displacement

of

the

sense

probe

induced

by

the

fluid

jet

was

measured

as

a

function

of

stimulus

frequency

The

frequency-dependent

amplitude

and

phase

characteristics

obtained

this

way

are

however

contaminated

with

frequency-dependent

characteristics

of

the

sense

probe

and

other

possible

frequency-selective

components

of

the

equipment

To

isolate

the

fluid

jet

frequency

response

three

additional

frequency

responses

were

measured

in

this

procedure

which

are

displayed

in

Figure

FIG.

Schematic

representation

of

the

fluid

jet

correction

procedure

A

Each

column

-LRB-

-RRB-

represents

a

type

of

measurement

needed

to

complete

the

correction

The

frequency

responses

measured

with

the

sense

probe

contain

frequency-dependent

properties

of

ED

-LRB-

equipment

demodulator

-RRB-

S

-LRB-

stimulus

sphere

-RRB-

SP

-LRB-

sense

probe

-RRB-

or

FJ

-LRB-

fluid

jet

-RRB-

as

a

function

of

frequency

Results

represented

in

a

column

are

used

to

correct

the

measured

response

in

the

next

column

indicated

by

the

long

arrows

and

produce

results

pointed

at

with

a

short

arrow

B

Displacement

amplitude

and

phase

for

each

measurement

-LRB-

-RRB-

described

in

-LRB-

A

-RRB-

Each

solid

line

is

the

result

of

a

correction

-LRB-

except

for

column

-RRB-

and

is

used

in

the

next

column

to

correct

the

measured

frequency

response

-LRB-

symbols

-RRB-

both

for

the

amplitude

and

the

phase

response

The

first

step

-LRB-

Fig.

-RRB-

is

to

obtain

the

frequency-dependent

properties

of

the

demodulator

-LRB-

D

-RRB-

including

the

additional

signal

measuring

and

conditioning

equipment

-LRB-

E

-RRB-

like

-LRB-

anti-alias

-RRB-

filters

and

amplifiers

For

this

step

a

voltage-controlled

oscillator

-LRB-

VCO

-RRB-

was

used

to

generate

a

frequency-modulated

kHz

carrier

signal

which

was

sent

to

the

demodulator

These

signals

of

the

VCO

at

various

known

modulation

frequencies

simulate

a

calibrated

output

of

the

photomultiplier

similar

to

operating

the

laser

interferometer

The

resulting

response

ED

-LRB-

f

-RRB-

characterizing

demodulator

-LRB-

D

-RRB-

and

equipment

-LRB-

E

-RRB-

is

presented

in

Figure

as

a

function

of

frequency

f

It

shows

a

nearly

constant

amplitude

as

a

function

of

frequency

and

a

clear

phase

change

starting

at

about

Hz

as

a

result

of

the

used

anti-alias

filter

The

second

step

characterizes

the

frequency

response

of

the

stimulus

sphere

S

-LRB-

f

-RRB-

The

laser

interferometer

is

used

to

directly

measure

its

displacement

amplitude

and

phase

as

a

function

of

input

frequency

with

fixed

voltage

amplitude

This

response

consisting

of

ED

-LRB-

f

-RRB-

S

-LRB-

f

-RRB-

-LRB-

Fig.

data

points

-RRB-

can

now

be

divided

by

ED

-LRB-

f

-RRB-

obtained

in

step

to

isolate

the

pure

stimulus

sphere

response

S

-LRB-

f

-RRB-

-LRB-

Fig.

solid

line

-RRB-

The

resonance

at

approximately

Hz

originating

from

the

mass

of

the

stimulus

sphere

and

its

elastic

attachment

can

clearly

be

discerned

The

fluid

displacement

produced

by

a

vibrating

stimulus

sphere

can

be

taken

proportional

to

the

displacement

of

the

sphere

S

-LRB-

f

-RRB-

-LRB-

e.g.

van

Netten

-RRB-

In

the

third

step

the

same

stimulus

sphere

vibrating

in

water

is

used

to

hydrodynamically

displace

the

sense

probe

which

has

frequency

characteristics

SP

-LRB-

f

-RRB-

that

subsequently

will

be

used

to

probe

the

fluid

jet

displacement

The

sense

probe

displacement

resulting

from

sphere

stimulation

ED

-LRB-

f

-RRB-

S

-LRB-

f

-RRB-

SP

-LRB-

f

-RRB-

as

measured

by

the

laser

interferometer

is

thus

contaminated

with

equipment

i.e.

demodulator

and

stimulus

sphere

characteristics

This

response

is

divided

by

the

previously

-LRB-

step

-RRB-

isolated

stimulus

sphere

response

S

-LRB-

f

-RRB-

resulting

in

the

sense

probe

frequency

response

ED

-LRB-

f

-RRB-

SP

-LRB-

f

-RRB-

-LRB-

Fig.

solid

line

-RRB-

which

is

intentionally

still

contaminated

with

the

equipment

response

characteristics

ED

-LRB-

f

-RRB-

In

the

fourth

step

this

result

ED

-LRB-

f

-RRB-

SP

-LRB-

f

-RRB-

is

used

to

divide

the

fluid

jet-driven

sense

probe

response

measured

by

the

interferometer

ED

-LRB-

f

-RRB-

FJ

-LRB-

f

-RRB-

SP

-LRB-

f

-RRB-

thus

eventually

isolating

the

amplitude

and

phase

of

the

pure

fluid

jet

displacement

response

FJ

-LRB-

f

-RRB-

as

a

function

of

the

frequency

-LRB-

Fig.

solid

line

-RRB-

Effects

of

tip

diameter

Measurements

The

dimensions

of

the

object

that

has

to

be

stimulated

largely

determine

the

diameter

of

the

tip

of

the

glass

pipette

to

be

used

for

producing

the

fluid

jet

Mechanically

stimulating

an

individual

hair

bundle

usually

requires

pipette

tip

-LRB-

i.e.

orifice

-RRB-

diameters

smaller

than

μm

whereas

some

tectorial

structures

like

a

lateral

line

neuromast

may

require

tip

openings

of

up

to

mm

A

narrow

tip

increases

the

outflow

resistance

and

thus

decreases

the

fluid

flow

of

the

jet

as

evidenced

by

the

larger

voltage

amplitudes

that

have

to

be

applied

across

the

piezoelectric

disc

to

obtain

equal

displacement

amplitudes

of

the

sense

probe

Another

important

consequence

of

the

tip

diameter

is

its

influence

on

the

fluid

flow

frequency

characteristics

of

the

fluid

jet

To

illustrate

this

calibrated

fluid

jet

characteristics

obtained

as

described

in

the

previous

section

are

shown

for

different

tip

diameters

in

Figure

This

figure

also

shows

the

amplitude

and

phase

response

of

the

resulting

fluid

displacement

when

using

a

pipette

without

a

tip

restriction

-LRB-

Fig.

-RRB-

In

this

case

the

glass

pipette

was

cut

at

the

same

length

as

the

pipettes

used

in

Figure

and

C

but

was

not

tapered

at

its

tip

Its

outflow

diameter

therefore

equals

the

inner

diameter

of

the

glass

pipette

-LRB-

mm

-RRB-

The

characteristics

of

the

fluid

jet

generated

using

such

an

unrestricted

tip

in

response

to

signals

applied

to

the

piezoelectric

disc

within

a

range

of

frequencies

and

with

a

fixed

amplitude

have

a

constant

displacement

amplitude

up

to

a

certain

frequency

-LRB-

about

Hz

-RRB-

and

the

jet

thus

effectively

behaves

as

a

displacement

stimulus

in

this

frequency

range

-LRB-

Fig.

data

points

-RRB-

At

higher

frequencies

the

amplitude

starts

increasing

and

approaches

a

resonance

at

about

Hz

Beyond

this

resonant

frequency

the

displacement

amplitude

decreases

at

a

rate

of

about

dB/dec

which

is

reminiscent

of

a

second-order

system

driven

beyond

resonance

Around

the

resonant

frequency

the

phase

rotates

over

an

angle

of

which

is

consistent

with

these

second-order

characteristics

FIG.

Effect

of

tip

diameter

on

fluid

jet

response

Data

points

are

the

measured

amplitude

and

phase

responses

as

a

function

of

frequency

using

a

pipette

without

a

tip

restriction

-LRB-

A

-RRB-

a

tip

-LRB-

B

-RRB-

or

a

tip

-LRB-

C

-RRB-

The

solid

lines

are

fits

to

the

data

with

the

model

description

-LRB-

Eqs

and

-RRB-

of

the

jet-producing

device

-LRB-

Fig.

-RRB-

Fixed

parameters

based

on

physical

sizes

and

properties

L

mm

L

mm

D

mm

D

mm

ρ

Fixed

parameters

S

piezo

N/m

R

c

Ns/m

Varied

parameters

A

S

p

N/m

R

p

Ns/m

F

N

-LRB-

B

-RRB-

S

p

N/m

R

p

Ns/m

F

N

-LRB-

C

-RRB-

S

p

N/m

R

p

Ns/m

F

N

A

different

jet

response

is

produced

with

a

pipette

with

an

intermediate

tip

restriction

-LRB-

μm

Fig.

data

points

-RRB-

At

low

frequencies

the

jet

's

fluid

displacement

is

proportional

to

the

voltage

applied

to

the

piezoelectric

disc

as

evidenced

by

the

constant

amplitude

and

the

almost

zero

phase

With

this

more

restricted

pipette

tip

the

displacement

amplitude

however

starts

declining

beyond

Hz

changing

towards

a

slope

of

about

dB/dec

with

an

according

phase

rotation

in

the

mid-frequency

range

At

about

Hz

the

slope

of

the

amplitude

starts

to

decrease

further

and

also

the

phase

lag

changes

to

values

more

negative

than

With

this

more

restricted

tip

diameter

-LRB-

μm

-RRB-

no

clear

resonance

is

evident

as

was

the

case

with

the

larger

diameter

-LRB-

mm

-RRB-

In

the

case

of

an

extremely

restricted

tip

with

a

diameter

of

μm

the

fluid

jet

device

produces

a

displacement

amplitude

response

which

declines

with

about

dB/dec

over

the

measured

frequency

range

-LRB-

Hz

Fig.

data

points

-RRB-

so

that

effectively

the

velocity

amplitude

of

the

fluid

jet

is

constant

within

this

frequency

range

Consistent

with

these

characteristics

is

the

almost

constant

phase

lag

of

except

at

the

higher

frequencies

where

it

starts

to

decline

further

These

frequency

characteristics

reflect

an

overdamped

system

In

the

case

of

a

small

pipette

tip

diameter

therefore

the

resulting

dynamics

lead

to

a

fluid

jet

of

which

the

velocity

is

proportional

and

in

phase

with

the

voltage

applied

to

the

piezoelectric

disc

within

the

measured

frequency

range

Model

calculations

To

get

a

better

understanding

of

the

physical

factors

influencing

the

output

characteristic

of

the

jet-producing

device

the

measured

data

were

described

by

a

mechanical

model

of

the

masses

of

the

fluids

related

to

the

two

main

compartments

of

the

device

i.e.

the

Perspex

container

part

and

the

pipette

-LRB-

see

Materials

and

Methods

-RRB-

The

mass

of

the

piezoelectric

disc

-LRB-

Mpiezo

-RRB-

is

combined

with

the

fluid

mass

contained

in

the

Perspex

fluid

container

-LRB-

Mc

-RRB-

because

they

can

be

considered

to

move

in

unison

having

a

total

mass

The

fluid

mass

in

the

glass

pipette

is

indicated

with

Mp

These

two

effective

masses

Mtot

and

Mp

are

mechanically

coupled

by

a

lever

and

a

spring

Sp

-LRB-

Fig.

-RRB-

The

displacement

amplitude

of

the

mass

in

the

pipette

Xp

is

considered

to

be

proportional

to

the

displacement

of

the

fluid

jet

emerging

from

the

pipette

tip

and

is

calculated

as

a

function

of

frequency

of

the

force

produced

by

the

piezoelectric

disc

The

force

is

assumed

to

be

proportional

to

the

electrical

signal

applied

to

the

piezoelectric

disc

The

results

of

this

model

were

subsequently

fitted

to

the

measured

data

points

-LRB-

Figs.

solid

lines

-RRB-

The

model

results

support

the

measurements

in

showing

that

the

value

of

the

resistance

controlled

by

the

tip

of

the

pipette

Rp

strongly

influences

the

fluid

jet

response

characteristics

When

fitting

the

model

to

the

data

obtained

for

different

tip

diameters

the

value

of

the

resistance

Rp

increases

more

than

two

orders

of

magnitude

-LRB-

Fig.

solid

lines

Rp

small

Ns/m

Figure

solid

lines

Rp

large

Ns/m

-RRB-

when

the

pipette

without

tip

restriction

-LRB-

Ølarge

mm

-RRB-

is

compared

with

the

most

restricted

pipette

-LRB-

Øsmall

μm

-RRB-

Fitting

the

flow

produced

by

the

pipette

tip

with

intermediate

opening

-LRB-

Ø

μm

-RRB-

results

in

an

intermediate

resistance

value

-LRB-

Fig.

solid

lines

Rp

Ns/m

-RRB-

Besides

these

expected

changes

in

Rp

the

value

of

Sp

needed

to

be

modified

-LRB-

to

N/m

-RRB-

for

obtaining

proper

model

fits

to

the

data

The

cone

washers

-LRB-

Fig.

denoted

with

W

-RRB-

are

the

most

compliant

structures

in

the

assembly

to

which

part

of

the

elastic

stiffness

Sp

can

most

likely

be

attributed

The

variation

in

Sp

found

between

individual

fluid

jets

could

be

due

to

the

variability

with

which

the

cone

washers

were

positioned

during

the

insertion

of

the

pipette

and

subsequently

tightened

In

addition

we

can

not

exclude

the

variability

in

the

amount

of

gas

or

air

bubbles

present

in

the

silicone

fluid

causing

these

variations

in

Sp

between

individual

fluid

jets

Except

for

Sp

and

Rp

all

other

parameters

of

the

model

were

kept

fixed

in

fitting

the

results

with

different

pipette

tip

dimensions

-LRB-

Fig.

-RRB-

Resonance

frequencies

Unless

overdamped

by

resistive

elements

a

combination

of

a

spring

and

a

mass

leads

to

a

resonance

phenomenon

and

is

accompanied

by

a

phase

rotation

and

an

amplitude

response

slope

change

of

dB/dec

around

the

resonance

frequency

The

present

model

including

two

sets

of

masses

and

elastic

elements

therefore

predicts

two

possible

resonant

frequencies

fl

and

fh

-LRB-

see

Materials

and

Methods

-RRB-

The

high

resonance

frequency

fh

is

beyond

the

calibrated

frequency

range

considered

-LRB-

kHz

-RRB-

This

resonance

is

related

to

the

summed

mass

-LRB-

Mtot

-RRB-

of

the

fluid

in

the

Perspex

container

-LRB-

Mc

-RRB-

and

that

of

the

piezoelectric

disc

-LRB-

Mpiezo

-RRB-

combined

with

the

parallel

stiffness

-LRB-

Spiezo

-RRB-

of

the

piezoelectric

disc

and

the

up-transformed

stiffness

of

the

cone

washers

-LRB-

-RRB-

as

effectively

acting

on

the

fluid

mass

in

the

Perspex

container

-LRB-

Fig.

Eq

-RRB-

The

existence

of

a

second

resonance

at

higher

frequencies

was

confirmed

by

applying

both

sinusoidal

signals

and

voltage

steps

to

the

piezoelectric

disc

of

the

fluid

jet

device

The

displacement

of

a

-LRB-

uncalibrated

-RRB-

sense

probe

in

front

of

a

small

pipette

tip

-LRB-

μm

-RRB-

of

the

fluid

jet

device

driven

by

a

voltage

step

clearly

shows

a

resonance

at

about

kHz

-LRB-

Fig.

-RRB-

The

same

value

was

found

using

a

series

of

sinusoidal

voltages

applied

to

the

piezoelectric

disc

The

high-resonance

frequency

also

coincides

with

the

resonance

frequency

of

the

back

side

of

the

piezoelectric

disc

measured

directly

using

laser

interferometry

while

the

disc

was

being

clamped

and

operated

in

the

same

fluid

jet

device

-LRB-

not

shown

-RRB-

FIG.

High

frequency

fluid

jet

resonance

Response

of

the

fluid

jet

to

an

unfiltered

step

driver

voltage

applied

to

the

piezoelectric

disc

showing

a

high

resonance

frequency

f

h

kHz

-LRB-

solid

line

calibration

as

indicated

-RRB-

Parameters

used

in

the

model

-LRB-

dashed

line

-RRB-

S

piezo

N/m

R

c

Ns/m

S

p

N/m

R

p

Ns/m

F

N

In

the

case

of

the

unrestricted

pipette

with

small

damping

the

low

resonant

frequency

fl

was

observed

-LRB-

Fig.

-RRB-

together

with

the

accompanying

phase

rotation

within

the

measured

frequency

range

This

low-frequency

resonance

fl

-LRB-

see

Eq

-RRB-

can

be

attributed

to

the

mass

of

fluid

in

the

pipette

-LRB-

Mp

-RRB-

and

the

combined

series

stiffness

Sser

that

is

associated

with

the

compliant

cone

washers

or

air

bubbles

-LRB-

Fig.

C

-RRB-

and

the

down-transformed

piezo

stiffness

which

effectively

also

acts

on

Mp

The

resonance

frequency

resulting

from

substituting

the

fitted

parameters

of

Figure

in

Eq

yields

fl

Hz

similar

to

the

measured

value

-LRB-

Hz

-RRB-

Applications

Mechanical

excitation

of

sensory

hair

cells

using

the

fluid

jet

To

measure

transducer

currents

at

acoustic

frequencies

whole-cell

voltage-clamp

recordings

of

individual

outer

hair

cells

were

performed

in

acutely

prepared

organs

of

Corti

of

mice

-LRB-

e.g.

Géléoc

et

al.

van

Netten

et

al.

-RRB-

The

hair

bundles

were

stimulated

by

the

present

fluid

jet

device

using

narrow

tips

-LRB-

Ø

μm

-RRB-

so

that

fluid

jet

velocities

were

proportional

to

the

voltage

applied

to

piezoelectric

disc

-LRB-

e.g.

Fig.

-RRB-

The

pipette

's

tip

was

aimed

at

the

hair

bundle

the

mechano-receptive

structure

of

a

hair

cell

at

an

angle

of

about

with

the

apical

plate

of

the

hair

cell

and

at

a

distance

of

approximately

μm

from

the

hair

bundle

Figure

shows

examples

of

inward

transducer

currents

measured

in

an

apical

mouse

outer

hair

cell

The

hair

bundle

was

excited

with

periods

of

a

pure

sinusoidal

fluid

stimulus

at

Hz

and

at

Hz

as

produced

by

the

micro-fluid

jet

device

The

driver

voltage

and

resulting

jet

velocity

are

symmetric

whereas

the

measured

transducer

currents

-LRB-

Fig.

Hz

Figure

Hz

-RRB-

are

asymmetric

This

is

expected

as

the

transducer

channels

only

gate

ionic

currents

when

they

are

deflected

in

the

direction

of

the

tallest

hair

bundles

-LRB-

e.g.

Hudspeth

Fettiplace

and

Ricci

-RRB-

FIG.

Application

of

the

fluid

jet

to

excite

sensory

hair

cells

A

Schematic

representation

for

recording

transducer

currents

in

cochlear

outer

hair

cells

Indicated

are

the

fluid

jet

's

pipette

tip

and

the

hair

cell

under

whole-cell

voltage-clamp

configuration

with

a

patch

pipette

at

the

lateral

side

The

sinusoidal

driver

voltage

-LRB-

DV

-RRB-

applied

to

the

piezoelectric

disc

is

indicated

Positive

voltages

correspond

with

the

fluid

moving

out

of

the

chamber

towards

the

hair

bundle

so

that

it

bends

in

the

excitatory

direction

towards

the

tallest

stereocilia

B

Transducer

currents

in

response

to

a

fluid

stimulus

at

Hz

-LRB-

C

-RRB-

Transducer

currents

in

response

to

a

fluid

stimulus

at

Hz

from

the

same

cell

and

using

the

same

fluid-jet

pipette

as

in

B.

D

Responses

of

transducer

currents

-LRB-

dashed

Hz

solid

Hz

-RRB-

shown

on

an

extended

time

axis

in

relation

to

two

stimulus

periods

of

the

driver

voltage

-LRB-

DV

thin

black

line

upper

trace

-RRB-

The

two

transducer

currents

have

a

comparable

magnitude

but

show

a

different

delay

in

activation

with

respect

to

the

maxima

of

the

driver

voltage

-LRB-

vertical

thin

lines

-RRB-

which

is

caused

by

the

different

phase

delays

of

the

fluid

jet

at

and

Hz

The

measured

delays

are

and

periods

for

respectively

the

Hz

and

the

Hz

stimulus

and

correspond

to

a

phase

delay

of

and

respectively

E

Step

driver

voltage

-LRB-

DV

-RRB-

applied

to

the

piezoelectric

disc

of

the

device

and

the

evoked

transducer

current

showing

a

oscillation

produced

by

resonance

of

the

fluid

in

the

Perspex

container

In

Figure

two

periods

of

the

traces

from

Figure

and

C

are

shown

on

an

extended

time

scale

For

both

frequencies

there

are

delays

in

peak

transducer

current

with

respect

to

the

maximum

stimulus

-LRB-

vertical

lines

-RRB-

The

measured

delays

are

μs

and

μs

-LRB-

mean

SD

n

-RRB-

for

respectively

the

Hz

and

the

Hz

stimulus

This

corresponds

to

a

phase

delay

of

at

Hz

and

to

at

Hz

which

can

be

attributed

to

the

frequency-dependent

phase

delay

of

the

fluid

jet

-LRB-

Fig.

see

also

Discussion

-RRB-

Hair

cells

also

reliably

transduce

the

-LRB-

high-frequency

-RRB-

resonance

of

the

fluid

jet

device

in

response

to

an

unfiltered

step

applied

to

the

piezoelectric

disc

as

can

be

seen

in

Figure

The

hair

cell

response

shows

an

initial

step

transducer

current

followed

by

an

oscillation

of

kHz

This

frequency

is

identical

to

the

resonance

observed

while

applying

an

unfiltered

fluid

jet

stimulus

to

the

sense

probe

-LRB-

Fig.

-RRB-

and

thus

illustrates

the

need

for

proper

filtering

of

the

voltage

step

to

avoid

fluid

jet

resonance

DISCUSSION

Reliability

of

the

calibration

method

The

flow

of

microscopic

fluid

jets

produced

by

several

types

of

devices

has

been

used

in

hair

cell

research

for

several

decades

-LRB-

Flock

and

Orman

Saunders

and

Szymko

Kros

et

al.

Rüsch

et

al.

Géléoc

et

al.

Vollrath

and

Eatock

van

Netten

et

al.

-RRB-

In

almost

all

cases

the

mechanical

output

characteristics

are

assumed

to

be

constant

in

terms

of

fluid

displacement

or

velocity

but

generally

lack

detailed

data

on

these

actual

output

characteristics

In

the

present

study

a

method

was

developed

to

dynamically

calibrate

a

microscopic

fluid

jet

using

a

flexible

sense

probe

that

was

independently

calibrated

by

means

of

an

additional

stimulus

device

-LRB-

stimulus

sphere

-RRB-

providing

its

detailed

frequency

selectivity

within

a

relevant

frequency

range

This

information

is

necessary

to

properly

analyze

the

displacements

of

objects

induced

by

a

fluid

jet

Because

changes

in

the

tip

diameter

of

the

jet-producing

pipette

alter

the

frequency

response

a

calibration

for

each

individual

pipette

is

necessary

especially

when

applied

at

higher

frequencies

-LRB-

e.g.

Fig.

-RRB-

The

developed

calibration

procedure

using

a

sense

probe

has

been

shown

to

be

practical

within

the

frequency

range

tested

-LRB-

Hz

-RRB-

This

calibration

procedure

intrinsically

produces

results

for

the

flow

of

the

jet

in

a

free

and

unobstructed

fluid

field

This

is

related

to

the

principle

used

which

is

based

on

equivalent

forces

acting

on

a

sense

probe

when

stimulated

with

either

a

fluid

jet

or

a

sphere

with

known

-LRB-

free

-RRB-

flow

field

The

frequency-dependent

characteristics

of

a

particular

sense

probe

can

be

determined

well

in

advance

of

the

actual

physiological

experiment

and

the

sense

probes

can

be

stored

for

later

use

During

an

experiment

a

sense

probe

can

be

positioned

in

close

proximity

of

the

preparation

to

enable

calibration

of

the

fluid

jet

directly

after

the

stimulation

of

the

preparation

without

having

to

move

the

pipette

tip

out

of

the

water

Essential

for

this

procedure

is

that

the

fluid

jet

's

frequency

response

does

not

change

over

time

A

potential

danger

is

that

the

pipette

tip

gets

-LRB-

partially

-RRB-

blocked

by

dirt

floating

in

the

bath

solution

Such

an

additional

restriction

of

the

tip

leads

to

changed

jet

response

characteristics

precluding

a

proper

correction

of

the

measured

data

However

in

practice

such

a

blockage

in

the

glass

pipette

's

tip

can

be

monitored

by

microscopic

visualization

Displacement

or

velocity

output

depends

on

tip

size

From

the

experiments

on

fluid

jets

produced

by

various

glass

pipettes

it

appears

that

the

pipette

's

tip

size

is

the

dominant

parameter

determining

the

fluid

jet

output

characteristics

both

in

terms

of

the

overall

output

and

frequency

selectivity

-LRB-

Fig.

-RRB-

One

of

the

most

relevant

features

of

this

stimulus

device

is

whether

it

is

the

fluid

displacement

or

velocity

of

the

jet

that

follows

the

voltage

applied

to

the

device

Our

results

show

that

with

a

large

tip

diameter

-LRB-

unrestricted

pipette

-RRB-

the

displacement

of

the

fluid

jet

is

proportional

to

the

displacement

of

the

piezoelectric

disc

in

a

low

frequency

range

-LRB-

Hz

-RRB-

This

displacement

mode

is

accompanied

with

resonance

behavior

-LRB-

Fig.

-RRB-

A

restriction

of

the

pipette

's

tip

diameter

critically

damps

the

system

preventing

it

from

resonating

Under

such

conditions

-LRB-

e.g.

Ø

μm

Fig.

-RRB-

it

produces

a

constant

fluid

displacement

amplitude

as

a

function

of

frequency

of

the

input

signal

up

to

about

Hz

Beyond

this

frequency

the

displacement

amplitude

falls

off

with

dB/dec

and

is

accompanied

by

a

phase

rotation

that

may

reach

at

high

frequencies

An

even

higher

restriction

of

the

tip

diameter

-LRB-

Ø

μm

Fig.

-RRB-

results

in

a

fluid

displacement

that

declines

with

about

dB/dec

from

low

-LRB-

Hz

-RRB-

to

high

-LRB-

Hz

-RRB-

frequencies

with

a

fixed

phase

delay

of

about

This

means

that

the

velocity

of

the

fluid

jet

follows

the

voltage

applied

to

the

piezoelectric

disc

This

is

the

most

effective

mode

for

producing

a

hydrodynamic

stimulus

to

excite

micro-sized

objects

such

as

sensory

hair

bundles

Comparison

of

measured

fluid

jet

characteristics

with

model

A

physical

representation

of

the

fluid

jet-producing

device

as

two

damped

fluid

masses

coupled

via

a

lever

and

loaded

with

springs

describes

the

behavior

of

the

fluid

jet

emerging

from

the

pipette

's

tip

quite

well

-LRB-

Fig.

solid

lines

-RRB-

It

follows

from

fitting

the

model

response

to

the

measured

data

that

the

dynamics

of

the

fluid

in

the

pipette

rather

than

that

in

the

larger

Perspex

container

dominantly

determines

the

overall

frequency

response

characteristics

below

kHz

In

this

frequency

range

the

output

can

be

approximated

with

Eq

which

shows

the

importance

of

Rp

and

Mp

in

relation

to

the

frequency

selectivity

It

also

shows

that

the

overall

output

at

frequencies

around

and

below

the

low

resonance

frequency

-LRB-

-RRB-

is

inversely

proportional

to

the

stiffness

of

the

piezoelectric

disc

The

high

resonance

frequency

of

the

fluid

jet

was

found

at

kHz

-LRB-

Fig.

-RRB-

The

value

of

this

resonance

frequency

was

confirmed

by

laser

interferometric

measurements

on

the

back

side

of

the

piezoelectric

disc

under

the

same

fluid

load

conditions

When

the

fluid

was

removed

from

the

Perspex

body

and

no

pipette

was

inserted

the

piezoelectric

disc

resonated

at

about

kHz

As

in

this

configuration

only

the

disc

is

moving

we

could

estimate

its

effective

dynamic

mass

value

in

combination

with

the

value

of

Spiezo

to

be

Mpiezo

kg

This

appears

to

be

of

the

physical

mass

of

the

disc

-LRB-

kg

-RRB-

which

seems

to

be

a

reasonable

fraction

if

the

circular

disc

is

moving

in

a

mode

with

a

circular

node

imposed

by

its

peripheral

clamping

support

The

compressibility

of

the

fluid

in

the

Perspex

container

and

that

in

the

pipette

has

been

neglected

This

can

be

justified

as

the

obtained

maximum

stiffness

values

of

the

elastic

elements

associated

with

these

masses

Spiezo

-LRB-

N/m

-RRB-

and

Sp

-LRB-

N/m

-RRB-

are

significantly

smaller

than

the

effective

stiffness

values

associated

with

their

compressibility

and

N/m

respectively

These

effective

stiffness

values

Seff

were

evaluated

by

considering

the

compressibility

along

the

long

axis

of

a

fluid-filled

cylinder

having

length

L

and

cross

sectional

area

A

and

can

be

shown

to

equal

Seff

KA

L

with

K

-LRB-

Pa

-RRB-

the

bulk

modulus

of

water

-LRB-

Lide

-RRB-

Highly

restricted

pipettes

with

tip

diameters

of

the

order

of

μm

or

less

strongly

overdamp

the

system

and

produce

a

velocity

output

which

is

proportional

to

the

force

produced

by

the

piezoelectric

disc

and

therefore

to

the

voltage

signal

applied

-LRB-

Fig.

-RRB-

The

model

therefore

predicts

the

transition

in

fluid

jet

characteristics

from

a

displacement

stimulus

in

the

situation

of

a

large

tip

diameter

-LRB-

small

tip

resistance

-RRB-

to

a

velocity

stimulus

due

to

tip

restriction

Figure

compares

these

model

results

in

more

detail

for

increasing

tip

resistance

Rp

At

a

relative

low

pipette

resistance

-LRB-

Rp

Ns/m

-RRB-

a

constant

displacement

amplitude

of

the

jet

is

produced

at

frequencies

below

about

Hz

showing

resonance

around

Hz

Increasing

the

resistance

to

Rp

Ns/m

critically

damps

the

system

making

it

a

useful

displacement

stimulus

device

up

to

about

Hz

Gradually

increasing

the

resistance

Rp

-LRB-

to

Ns/m

-RRB-

changes

the

fluid

jet

apparatus

into

a

velocity-producing

device

in

the

full

frequency

range

considered

-LRB-

Hz

-RRB-

at

the

expense

of

the

jet

's

flow

amplitude

This

velocity

output

corresponds

with

a

slope

of

dB/dec

in

the

displacement

amplitude

response

FIG.

The

effect

of

increasing

pipette

resistance

Series

of

modelled

frequency

responses

showing

fluid

displacement

output

as

a

function

of

the

frequency

of

the

voltage

signal

put

across

the

piezoelectric

disc

Parameter

values

L

mm

L

mm

D

mm

D

mm

ρ

S

piezo

N/m

R

c

Ns/m

S

p

N/m

values

of

pipette

resistance

-LRB-

R

p

in

Ns/m

-RRB-

are

given

next

to

each

curve

Absolute

fluid

jet

velocity

The

calibration

method

described

was

focused

on

the

relative

frequency

selectivity

of

the

fluid

jet

emerging

from

the

tip

of

the

glass

pipette

The

absolute

fluid

velocity

that

can

be

obtained

not

only

depends

on

the

tip

diameter

but

it

also

varies

strongly

with

the

measurement

position

with

respect

to

the

pipette

's

tip

This

means

that

it

is

not

useful

to

afterwards

calibrate

the

absolute

fluid

velocity

on

a

sense

probe

as

done

for

determining

the

relative

frequency

sensitivity

Nevertheless

an

estimate

of

the

order

of

magnitude

follows

directly

from

curves

like

Figure

or

alternatively

by

visualizing

the

trajectories

of

small

-LRB-

Ø

μm

-RRB-

latex

microbeads

in

the

flow

field

-LRB-

e.g.

Sauders

and

Szymko

-RRB-

Our

observations

for

instance

show

that

a

sensitivity

of

about

mm/s

per

volt

applied

to

the

piezoelectric

disc

is

obtained

at

a

distance

of

about

μm

in

front

of

the

pipette

tip

-LRB-

Ø

μm

-RRB-

From

such

calibrations

we

estimate

that

the

fluid

flow

velocity

required

to

produce

an

open

probability

of

the

transducer

channels

of

is

about

mm/s

Linearly

extrapolating

this

to

the

maximum

voltages

that

may

be

applied

to

the

piezo-electric

element

-LRB-

V

-RRB-

without

damaging

it

leads

to

a

maximal

velocity

in

the

order

of

mm/s

Applications

As

shown

in

Figure

a

fluid

jet

produced

by

the

apparatus

using

a

restricted

tip

diameter

-LRB-

μm

-RRB-

can

be

adequately

used

to

effectively

evoke

robust

transducer

currents

in

individual

sensory

hair

cells

It

is

known

from

previous

studies

-LRB-

van

Netten

et

al.

Ricci

et

al.

-RRB-

that

the

speed

of

the

transducer

apparatus

in

mammalian

outer

hair

cells

is

well

exceeding

the

range

of

frequencies

-LRB-

to

Hz

-RRB-

considered

here

This

implies

that

the

transducer

apparatus

is

capable

of

following

signals

up

into

the

kiloHertz

range

without

delay

The

observed

delays

in

transducer

currents

-LRB-

Fig.

-RRB-

can

therefore

be

attributed

to

the

frequency

selectivity

of

the

fluid

jet

Figure

illustrates

that

the

difference

in

phase

delay

at

Hz

-LRB-

-RRB-

compared

to

that

at

Hz

-LRB-

-RRB-

amounts

to

about

when

using

restricted

pipettes

The

observed

time

delays

of

the

hair

cell

responses

-LRB-

Fig.

-RRB-

in

stimulus

periods

-LRB-

at

Hz

and

at

Hz

-RRB-

or

in

terms

of

phase

delays

-LRB-

at

Hz

and

at

Hz

a

difference

of

-RRB-

are

therefore

in

line

with

the

measured

phase

difference

of

the

fluid

jet

at

these

frequencies

As

is

also

apparent

from

Figure

the

amplitude

response

of

the

fluid

jet

is

quite

comparable

in

terms

of

velocity

of

the

jet

at

and

Hz

in

line

with

the

equal

magnitude

of

the

measured

transducer

currents

at

these

frequencies

These

results

support

our

notion

that

a

proper

calibration

of

the

fluid

jet

characteristics

is

necessary

for

a

correct

interpretation

of

the

-LRB-

phase

-RRB-

measurements

A

relevant

issue

related

to

the

application

to

hair

cell

stimulation

is

the

proper

mechanical

stimulus

to

excite

a

sensory

hair

bundle

Based

on

the

micromechanics

and

hydrodynamics

of

an

individual

hair

bundle

it

has

theoretically

been

argued

that

a

hair

bundle

's

displacement

response

is

flat

as

a

function

of

the

frequency

of

a

fluid

velocity

stimulus

up

to

a

cut-off

frequency

determined

by

its

stiffness

and

its

size

-LRB-

Géléoc

et

al.

van

Netten

-RRB-

The

pipettes

used

to

displace

individual

bundles

have

opening

diameters

that

are

matched

to

the

dimensions

of

the

bundle

resulting

in

diameters

of

about

μm

or

below

When

applying

a

-LRB-

filtered

-RRB-

step

voltage

the

accompanying

high

resistance

of

the

pipette

tip

results

in

a

jet

velocity

proportional

to

the

step

and

therefore

produces

an

adequate

stimulus

to

statically

displace

individual

hair

cell

bundles

inducing

step

displacements

of

the

bundle

If

unfiltered

the

velocity

output

may

however

be

contaminated

with

a

ringing

-LRB-

kHz

-RRB-

caused

by

resonance

of

the

fluid

in

the

Perspex

fluid

container

-LRB-

Fig.

-RRB-

Adequate

low-pass

filtering

of

the

applied

voltage

signal

can

prevent

these

oscillations

but

will

inevitably

put

an

upper

limit

to

the

displacement

rise

time

of

a

hair

bundle

that

can

be

achieved

-LRB-

see

also

next

section

-RRB-

A

second

consequence

of

tip

restriction

is

the

decreased

output

response

Much

larger

voltages

have

to

be

applied

to

the

piezoelectric

disc

to

produce

similar

fluid

jet

displacements

This

consequence

of

tip

restriction

was

previously

recognized

by

Saunders

and

Szymko

-LRB-

-RRB-

who

showed

a

square

relation

between

fluid

displacement

amplitude

and

tip

diameter

Remarkably

they

did

not

report

differences

in

the

frequency

response

of

their

fluid

jet

due

to

tip

diameter

Their

calibrating

method

did

not

give

precise

information

about

the

phase

of

the

response

and

lacks

relevant

information

on

tip

size

Their

reported

displacement

frequency

responses

have

a

slope

of

dB/dec

which

makes

their

apparatus

neither

a

true

displacement

nor

a

true

velocity-producing

device

When

a

preparation

is

only

stimulated

at

a

single

frequency

the

detailed

frequency

response

of

the

fluid

jet

is

less

relevant

as

long

as

the

desired

displacement

and

related

phase

of

the

object

can

be

established

Harmonic

distortion

of

the

produced

fluid

jet

might

however

cause

a

problem

in

the

investigation

of

a

system

's

nonlinearities

Although

the

harmonics

produced

by

the

described

device

are

relatively

small

-LRB-

e.g.

Fig.

THD

dB

-RRB-

and

comparable

to

the

fluid

jet

distortion

reported

by

Saunders

and

Szymko

-LRB-

-RRB-

they

may

nevertheless

preclude

the

direct

measurement

of

transducer-related

hair

bundle

nonlinearities

from

harmonic

analyses

as

predicted

for

specific

types

of

hair

cells

-LRB-

e.g.

van

Netten

and

Kros

-RRB-

Bandwidth

limitations

of

the

fluid

jet-producing

device

The

calibration

method

described

here

is

restricted

to

about

Hz

because

of

the

frequency

range

of

the

stimulus

sphere

used

to

calibrate

the

sense

probes

which

resonates

at

about

Hz

and

produces

only

a

limited

fluid

displacement

beyond

resonance

-LRB-

Fig.

-RRB-

However

it

may

be

possible

to

design

a

method

to

mechanically

calibrate

the

sense

probes

into

the

supra-kHz

range

so

that

the

jet

's

output

can

also

be

calibrated

in

an

extended

frequency

bandwidth

In

the

velocity-producing

mode

resonance

of

the

fluid

mass

Mc

in

the

Perspex

container

may

limit

the

use

of

the

device

beyond

the

high

resonance

frequency

-LRB-

kHz

-RRB-

where

the

fluid

jet

velocity

output

declines

with

dB

dec.

It

may

seem

therefore

that

a

fundamental

restriction

of

the

application

of

the

fluid

jet

produced

by

the

apparatus

described

lies

in

the

restricted

bandwidth

imposed

by

the

piezoelectric

disc

used

The

piezoelectric

disc

's

intrinsic

resonance

behavior

under

dry

unloaded

conditions

i.e.

while

being

clamped

at

its

rim

similarly

as

in

the

operational

fluid

jet

device

but

without

the

fluid

load

imposed

is

about

kHz

However

using

the

model

it

can

be

shown

that

this

is

not

a

fundamental

limit

for

its

dynamic

use

as

the

high-frequency

resonance

is

also

determined

by

the

stiffness

Sp

which

contributes

via

up-transforming

via

the

lever

with

a

factor

to

the

stiffness

of

the

moving

fluid

in

the

Perspex

container

determining

the

high-resonance

frequency

-LRB-

Eq

-RRB-

Enhanced

-LRB-

high

-RRB-

frequency

characteristics

i.e.

a

higher

resonance

or

cut-off

frequency

-LRB-

fh

-RRB-

in

the

velocity

mode

using

small

pipette

tip

diameters

may

thus

be

expected

from

designs

with

a

smaller

fluid

mass

-LRB-

Mc

-RRB-

in

the

Perspex

container

improved

lever

action

-LRB-

higher

L

-RRB-

and

overall

stiffer

elastic

elements

-LRB-

Spiezo

Sp

-RRB-

although

the

latter

will

decrease

the

overall

output

of

such

devices

In

practice

this

means

that

at

the

expense

of

a

smaller

output

faster

fluid

jet

performance

may

be

obtained

by

using

smaller

but

stiffer

-LRB-

ultrasonic

-RRB-

piezo

elements

Conclusions

This

work

emphasizes

the

importance

of

the

calibration

of

a

jet-producing

stimulus

device

and

demonstrates

the

importance

of

the

outflow

restriction

for

the

jet

response

characteristics

Not

only

the

jet

output

is

decreased

with

a

higher

tip

resistance

but

more

importantly

its

output

characteristic

changes

from

a

displacement

to

a

velocity

fluid

stimulus

in

relation

to

the

input

voltage

of

the

device

A

simple

mechanical

model

that

adequately

describes

the

fluid

jet

's

characteristics

can

help

understanding

the

physical

parameters

underlying

these

characteristics

and

can

thus

facilitate

optimizing

the

design

of

this

type

of

hydrodynamic

stimulus

device

for

particular

applications

micro fluid jet
hydrodynamic excitation
calibrated fluid sensing