Diffusion - Adsorption Afﬁnity sensors · Course program • Afﬁnity sensors - Immunoassay - Protein chip • Biosensors - Coulometric & Amperometric Glucose sensors, Activated

Diffusion - AdsorptionAffinity sensors

Hubert H. Girault - Niels LionSpring 2011

Tuesday, March 1, 2011

Course program

• Affinity sensors - Immunoassay - Protein chip

• Biosensors - Coulometric & Amperometric Glucose sensors, Activated Clotting Time

Dr. Niels Lion

• Gel electrophoresis and Proteomics

• Mass spectrometry and proteomics methods

• Case study


Course timetableBiosensors -2011

• 24 - 2 : Affinity sensors - Diffusion adsorption

• 3 - 3 : Immunoassay - ELISA

• 10- 3 : Immunoassay - ELISA

• 17 - 3 : Glucose - Coulometry

• 24 - 3 : Dr. Joël Rossier - DiagnoSwiss

• 31 - 3 : Glucose - Amperometry

• 7 - 4 : Activated Clotting Time


Biosensor Project

• Answer the questions of the lecture notes

• Write a report on pregnancy test including a description of the device with cost and distribution network, its condition of use, the physico-chemical and biological principles, a market survey of the key players.

http://hcgpregnancytest.com/


http://hcgpregnancytest.com

http://hcgpregnancytest.com

Diagnostics


Biosensor definition

A reaction between the immobilized agent and the molecule being analysed is transduced into an electric signal.

A device that uses an immobilized agent (such as an enzyme, antibiotic, organelle or whole cell) to detect or measure a chemical compound.

(+ substrate) + Transducer

Receptor


How to detect a molecule on a surface?

•Label - free techniques

Conductivity sensors - NanowiresCapacitance sensors Surface plasmon resonanceMechanical sensors - QCM - ResonatorOptical grating

•Labelling techniques

DNA chipsProtein arraysImmunoassays


Direct Ultrasensitive Electrical Detection

of DNA and DNA Sequence VariationsUsing Nanowire NanosensorsJong-in Hahm and Charles M. Lieber*

Department of Chemistry and Chemical Biology, HarVard UniVersity,12 Oxford Street, Cambridge, Massachusetts 02138

Received October 2, 2003; Revised Manuscript Received November 10, 2003

ABSTRACT

The development of electrically addressable, label-free detectors for DNA and other biological macromolecules has the potential to impact

basic biological research as well as screening in medical and bioterrorism applications. Here we report two-terminal silicon nanowire electronicdevices that function as ultrasensitive and selective detectors of DNA. The surfaces of the silicon nanowire devices were modified with

peptide nucleic acid receptors designed to recognize wild type versus the !F508 mutation site in the cystic fibrosis transmembrane receptor

gene. Conductance measurements made while sequentially introducing wild type or mutant DNA samples exhibit a time-dependent conductanceincrease consistent with the PNA!DNA hybridization and enabled identification of fully complementary versus mismatched DNA samples.

Concentration-dependent measurements show that detection can be carried out to at least the tens of femtomolar range. This nanowire-based

approach represents a step forward for direct, label-free DNA detection with extreme sensitivity and good selectivity, and could provide apathway to integrated, high-throughput, multiplexed DNA detection for genetic screening and biothreat detection.

The development of advanced biological sensors could

impact significantly the areas of genomics,1,2 proteomics,3

biomedical diagnostics,4 and drug discovery.5 In this regard,

nanoscale sensors based on nanowires (NWs),6 nanotubes

(NTs),7 and other nanomaterials8-11 have received consider-

able recent attention. Nanoparticle labels have been used to

enhance the sensitivity in surface plasmon resonance (SPR)-

based detection,9,11 including DNA sensing.11 Metal submi-

cron wire barcodes also have been used in conjunction with

traditional fluorescent assays for DNA detection, where the

barcode makes possible multiplexed detection.10 In contrast

to these optical methods, NWs and NTs can be used for label-

free, direct real-time electrical detection of biomolecule

binding.6,7 NWs and NTs also have the potential for very

high sensitivity detection since the depletion or accumulation

of charge carriers, which are caused by binding of a charged

biological macromolecule at the surface, can affect the entire

cross-sectional conduction pathway of these nanostructures.

While both silicon NWs (SiNWs) and NTs have been used

previously for detecting biological species,6,7 we have focused

our efforts on SiNWs, since the electrical properties and

sensitivity of SiNWs can be tuned reproducibly by control-

ling dopant concentration and NW diameter.12-14 The

modification of silicon oxide surfaces also has been well

studied,15 and this information can be exploited for tailoring

SiNW surfaces with biological or chemical receptors.

Herein, we report the use of SiNWs for real-time, label-

free detection of DNA and DNA mismatches. We show that

SiNW sensors functionalized with peptide nucleic acid (PNA)

receptors can distinguish wild-type from the !F508 mutationsite in the cystic fibrosis transmembrane receptor (CFTR)

gene. Cystic fibrosis is one of the most common fatal genetic

diseases among populations of European origin and affects

Figure 1. (A) Schematic of a sensor device consisting of a SiNW(yellow) and a microfluidic channel (green), where the arrowsindicate the direction of sample flow. (B) The SiNW surface withPNA receptor. (C) PNA-DNA duplex formation.

NANO

LETTERS

2004Vol. 4, No. 151-54

10.1021/nl034853b CCC: $27.50 © 2004 American Chemical SocietyPublished on Web 12/09/2003





ABSTRACT








































NANO

LETTERS

2004Vol. 4, No. 151-54






ABSTRACT








































NANO

LETTERS

2004Vol. 4, No. 151-54


The development of electrically addressable, label-free detectors for DNA and other biological macromolecules has the potential to impact basic biological research as well as screening in medical and bioterrorism applications. Here we report two-terminal silicon nanowire electronic devices that function as ultrasensitive and selective detectors of DNA. The surfaces of the silicon nanowire devices were modified with peptide nucleic acid receptors designed to recognize wild type versus the ∆F508 mutation site in the cystic fibrosis transmembrane receptor gene. Conductance measurements made while sequentially introducing wild type or mutant DNA samples exhibit a time-dependent conductance increase consistent with the PNA−DNA hybridization and enabled identification of fully complementary versus mismatched DNA samples. Concentration-dependent measurements show that detection can be carried out to at least the tens of femtomolar range. This nanowire-based approach represents a step forward for direct, label-free DNA detection with extreme sensitivity and good selectivity, and could provide a pathway to integrated, high-throughput, multiplexed DNA detection for genetic screening and biothreat detection.


Biacore - SPR

Surface Plasmon Resonance to measure the refractive index above a gold film


Biacore-Label free detection


Biacore - Sensorgram


Surface Plasmon Resonance


Adsorption based sensors

Copyright © 2010 General Electric Company

Investigate interactions with small peptides through to multiple sub-unit protein complexes.Study native, recombinant, synthetic or tagged recombinant molecules.Discover new interaction partners in body fluids, cell culture supernatants or crude extracts.

Study the interaction of small molecules, such as drug candidates, with their targets.

Study membrane biochemistry or membrane-bound receptor interactions using native membranes, artificial membranes or vesicles.

Investigate replication, transcription and translation. Determine molecular relationships during the formation of protein complexes and their interaction with DNA.Study hybridization of DNA and RNA.

Study interactions involving whole cells or viruses

Study the effects of glycosylation on molecular interactions.Determine specific recognition properties of cell surface carbohydrates.


Biorad SPR


The ProteOn XPR36 protein interaction array system is a surface plasmon resonance (SPR) optical biosensor, designed to provide all of the benefits of parallel processing. XPR technology, a unique approach to SPR multiplexing, greatly improves the efficiency and flexibility of your experimental design, enabling you to run more experiments in a shorter period of time. The ProteOn XPR36 workflow is guided by ProteOn Manager software — an easy-to-use and intuitive software package. ProteOn Manager software uses a flexible, guided approach to instrument control, experimental setup, and data analysis. The ProteOn XPR36 system includes all components for successful and efficient protein interaction analysis — instrumentation, software, sensor chips, buffers and reagents, and protocol development kits.

Key Features and Benefits

• 6 x 6 array — monitor up to 36 different interactions in a single run on a single chip• One-shot kinetics — perform a complete kinetic experiment in a single experimental "shot"• Easy-activation sensor chip chemistry — utilize the exceptional coupling efficiency and very high

ligand activity• Flexible experimental design — test a variety of experimental conditions in a single run• High sample throughput — immobilize a panel of 6 ligands, and screen multiple panels of analytes

(up to 180 interactions per hour)Applications

• Antibody characterization and development• Drug-target interactions• Protein interface analysis• Protein complexes and cascades

Type and Range of Results

• Specificity/identity — determines binding specificity of molecules (qualitative "yes/no" binding)• Binding kinetics — calculates the rates at which two molecules associate and dissociate (on-off

rates; association constant (ka); dissociation constant (kd))• Binding affinity — deciphers the strength of attraction between two molecules (equilibrium analysis

(KD))• Concentration determination — measures the concentration of proteins based on a standard curve


Analyst, TheDOI: 10.1039/b006221i

High sensitivity microgravimetric biosensor for qualitative and quantitative diagnostic detection of polychlorinated dibenzo-p-dioxins

Quartz CrystalMicrobalance


a, A suspended microchannel translates mass changes into changes in resonance frequency. Fluid continuously flows through the channel and delivers biomolecules, cells or synthetic particles. Sub-femtogram mass resolution is attained by shrinking the wall and fluid layer thickness to the micrometre scale and by packaging the cantilever under high vacuum.

b, While bound and unbound molecules both increase the mass of the channel, species that bind to the channel wall accumulate inside the device, and, as a result, their number can greatly exceed the number of free molecules in solution. This enables specific detection by way of immobilized receptors.

c, In another measurement mode, particles flow through the cantilever without binding to the surface, and the observed signal depends on the position of particles along the channel (insets 1–3). The exact mass excess of a particle can be quantified by the peak frequency shift induced at the apex.

Nature, Vol 446-2007| doi:10.1038/nature05741

Resonator


a, The 200 33 7 m (length width thickness) microcantilever containing a 3 8 m (height width) channel is suspended in a vacuum cavity (optical micrograph, right). Microfluidic bypass channels (30 100 m, height width) are connected to the inlet and the outlet of the suspended channel, and enable the quick exchange of samples by pressure driven flow (red arrows). The electron micrograph (left) shows a bottom view of a cantilever that has been intentionally etched open to visualize the fluidic conduit inside.

b, Frequency response plots of a cantilever before (blue) and after (red) filling with water reveal different resonance frequencies but indistinguishable quality factors. To measure the frequency response, we monitored the vibration amplitude with a laser and a position sensitive photodetector (PSD) while the cantilever was being driven electrostatically at different frequencies (inset; E denotes the electric field, and 'DC' represents a bias voltage of 60 V).



a, Antibodies to goat IgG are immobilized on the native silicon dioxide surface of the suspended microchannel resonator in three steps, as illustrated by the pictograms along the frequency trace: electrostatic adsorption of poly(ethyleneglycol)-biotin grafted poly-l-lysine (PLL-PEG-biotin, 1 mg ml-1), binding of Neutravidin (0.5 mg ml-1), and attachment of biotinylated antibodies (0.5 mg ml-1) to the Neutravidin. The mass increase at each step can be followed in real time (1Hz 300 fmol·cm-2 for 150 kDa IgG molecules). One bypass channel was continuously rinsed with phosphate buffered saline (PBS) while the other bypass contained the sample. During the injection interval (red), the bypass holding the sample was pressurized, and during rinse cycles (blue), the pressure difference was reversed, as shown in the inset at the bottom of the panel.

b, Goat anti-mouse IgG was injected at concentrations from 0.7 nM to 0.7 M (blue traces). Between measurements, the surface was regenerated by injecting 200 mM glycine (HCl; pH 2.5), which dissociated the analyte while preserving the activity of the antibodies. Control injections with no IgG (black) or human IgG (red) showed very low levels of non-specific binding.



a, Synthetic particles of known size and density were measured to calibrate the mass sensitivity of the device. Gold nanoparticles (100 8 nm) weighing 10 fg more than the water they displace produced a mean frequency shift of 36 mHz with a standard deviation of 6 mHz. On a different device, we measured a frequency shift of 310 30 mHz for polystyrene microspheres (1.51 0.01 m) with 90.1 fg mass excess.

b, The masses of E. coli and B. subtilis in PBS were measured by passing the bacteria through the resonator and collecting peak height histograms.



Zeptosens

The ZeptoREADER is based on a planar optical waveguide (PWG) technique. Light of a plane polarized laser is coupled into a waveguide with an optical grating. It then propagates via total internal reflection and produces an evanescent field which penetrates the vicinity of the waveguide up to 100 to 200 nm. Through a flow cell, the surface can be brought in contact with a liquid. The instrument is detecting fluorescence which is emitted when fluorescent dyes get excited by the evanescent field in the vicinity of the waveguide. It is therefore an in situ and highly surface sensitive technique.

http://www.zeptosens.com/en/




Enzyme Linked ImmunoSorbent Assay ELISA

Y Y Y Y

YY

YYY

Y

Capture antibody adsorptionWashSurface blockingWashAntigen adsorptionWashY Y

Y

Counter antibody adsorption

Addition of substrateDetection

A market of more than 20 billion $...

Total well volume = 300μL100 μL = 1cm2


Langmuir isotherm

kon koff

Surface bulk equilibriumNo lateral interactions


Statistical approach

Physica A 376 (2007) 27–37

Sample configurations of adsorbed hard spheres on the random site surface. A sphere may bind, centered, to a site as long as the nearest occupied site is at least σ away ( σ=sphere diameter). Left configuration: α=1;Number of point sites Ns=200; θ=0.380;Right configuration: α=10;Ns=2000; θ=0.650.In both cases λ=exp(μ/kT)= 10 000.

L


Adsorption reaction

Empty site + species Adsorbed specieskon

koff

: Surface concentration (mol·m–2 )

max : Maximum surface concentration (mol·m–2 )


Two layer model

Bulk

cbulk

Surface layer

zcsurf

Adsorbed layer


Langmuir isotherm

For antigen binding eq = maxKcbulk <<1

�

K =konkoff

=

max ( ) cbulk=

1( ) cbulk

=

max : Surface coverage (dimensionless)

Adsorption equilibrium constant

�

eq =Kcbulk

1+ Kcbulk=

1+

Equilibrium coverage


Example : IgG-Antigen

K=109 M–1

kon=106 M–1·s–1

cbulk = 0.1 pM

D = 10–10 m2·s–1

δ = 100 μm

Γmax = 10–9 mol·m–2

Site size : 40 nm x 40 nm

= 104

eq = 104

The equilibrium surface coverage for a 0.1 pM solution is 0.01% of a monolayer


Kinetically controlled adsorption

Bulk

Infinite mass transportLarge volume

csurf = cbulk


Adsorption kinetics

ddt

= koncbulk (1) koff

Differential equation

ddt

+ koncbulk + koff

= konc

bulk

Solution

=konc

bulk

koncbulk + koff

1 exp koncbulk + koff( )t

=

+11 exp t / [ ]

Time constant =1

koncbulk + koff

=td1+

time

1 >> 1

= 1

< 1

1/2 No mass transportlimitations


Characteristic times

Desorption time td = 1 / koff

Adsorption time ta = 1 / koncbulk

Time constant1

=1ta+1td

The smallest time determines the time constant


Time constant

5

4

3

2

1

0

-1

-2

log (

!

/ s

)

-15 -10 -5 0

log (c /M)

kon = 105

koff = 10–4

koff = 10–3

koff = 10–2

koff = 10–1

!

5

4

3

2

1

0

-1

-2lo

g (

!

/ s

)-15 -10 -5 0

log (c /M)

kon = 106, 10

5, 10

4, 10

3

koff = 10–4

!

Transition point : cbulk = K –1

td << ta td >> ta


Low concentrations

At low bulk concentrations << 1( )

At short times in general

= koncbulkt = t / ta

= eqttd

For dilute solutions, the kinetic is controlled by the desorption time

!(t) = !eq 1" exp "t / td( )#$ %&


Example : IgG-AntigenSite size : 40 nm x 40 nm

= 104

eq = 104

ta = 107s

td = = 103s

1.0

0.8

0.6

0.4

0.2

0.0

/ e

q

500040003000200010000

time/s

We need about one hour to reach 10–4 monolayer

K=109 M–1

kon=106 M–1·s–1

cbulk = 0.1 pM

D = 10–10 m2·s–1

δ = 100 μm



Rate constants

kon / M–1·s–1 koff / s

–1

IgE 106 10–3

IgG4 106 10–2

Avidin-Biotin 108 10–7

ss-DNA 6·104 5·10–5

Exercise : Please comment on the different values


Tyranny of the Langmuir isotherm

It takes a very long time to adsorb molecules on a surface from a dilute solution...


Describe a label-free technique

Draw the ideal sensorgram (infinite mass transport) for the adsorption of a species from a 1 nM solution

for 20 mn, and then the desorption

K=109 M–1

kon=106 M–1·s–1

cbulk = 1 nM


Adsorption Desorption

Time


Micro Immunoassay

Eight-by-eight íMIA representing the experiments and their outcome as revealed simultaneously in the immunofluorescence color channels. (A) Different species antibodies are immobilized as lines onto a PDMS substrate using a íFN (1). The BSA blocking step (2) is done before the cross-delivery of solutions providing the positive,negative controls, four samples, and FITC-labeled protein A (3). As all antibodies in the samples and protein A are labeled with either FITC or TRITC, it is possible to predict the mosaic pattern, represented by colored squares. “w” and “c” denote the expectation of weak binding and cross-reactivity, respectively, between the antigens and the delivered proteins. (B) The mosaic corresponds to the recom-bination of the FITC and TRITC channel images (left side) acquired with a fluorescence microscope. The mosaic is consistent with the diagram and reveals nonanticipated cross-reaction.

Anal. Chem. 2001, 73, 8-12Tuesday, March 1, 2011

Adsorption in a microchannel

t=0 c0

Equilibrium ce < c0

Adsorption



First fill nin = cinV

nin = nwall + nsolution = eqµ A + ceqV = KmaxceqA + ceqV

eqµ =

KmaxVV + KmaxA

�

��

�

��cin =

eq1+ KmaxA /V

=eq1+

The equilibrium surface concentration depends on the surface to volume ratio



=nwallnin

=KmaxAV

1

1+ KmaxAV1 =

11+

Collection efficiency

=h

Kmax=

VAKmax

Geometric factor

In a microsystem, the surface is large compared to the volume, and the geometric factor tends to zero

In a microsystem, the collection efficiency tends to unity

!eqµ =

! !eq1+!

" ! !eq



K=109 M–1

kon=106 M–1·s–1

cbulk = 0.1 pM

D=10–10 m2·s–1

δ = 100 μm


100μm

50μm

1cm

V = 50nL

A = 3mm2

= 0.01667 = 0.9836

eq 10–6

DiagnoSwiss Immunochip

With one fill, we have only 0.0001% of a monolayer for a 0.1pM solution


Kinetics in microsystems

Large volume ddt

= koncbulk (1) koff

Microsystems dnwallnsdt

= konnin ! nwall

V"#$

%&'ns ! nwall

ns

"#$

%&'! koff

nwallns

"#$

%&'

Wall coverage

nwall = nin1+! 1+ Kcin( ) 1! exp !

kon nin + ns( )V

+ koff"#$

%&'t

(

)*

+

,-

(

)**

+

,--

dnwalldt

+kon nin + ns( )

V+ koff

!"#

$%&nwall = konninns

V


Kinetics in microsystems

µ =td

1+ + –1Adsorption time constant

<< 1For dilute solutions:

In microsystems, the apparent desorption time is smaller

nwall = !nin 1! exp ! t!"td

"

#$

%

&'

"

#$

%

&'



K=109 M–1

kon=106 M–1·s–1

cbulk = 0.1 pM

D=10–10 m2·s–1

δ = 100 μm


100μm

50μm

1cm

All the molecules adsorbed in less than 2 minutes

1.0

0.8

0.6

0.4

0.2

0.0

/ e

q

10008006004002000

time/s

V = 50nL

A = 3mm2

= 0.01667 = 0.9836td = 16.4s

tdif = h2 / 2D = 3s


IgG adsorption on a PET microchannel

Confocal fluorescence detection of adsorption in a microchannel

chapter 2. Confocal Microscopy to Observe Proteins Adsorption in Microsystems 25

derivatives at 5 ! 10-12 M were detected in microtiter plates.4 Detection in gels and capillary

arrays was achieved by Mathies and co-workers.5, 6 The analysis of matrixes for

chromatography, the visualisation of protein adsorbed on different supports, the study of

diffusant in polymer films are other possibilities offered by this technique.

Over the last few years, our laboratory has investigated the mass transport properties of

proteins in microsystem. In particular, microimmunoassays in disposable polymer chips

integrating electrodes for electrochemical detection have been developed.7, 8

For the present work, there was the need to study the behaviour of proteins in polymer

supports used in microimmunoassays (behaviour that often means adsorption, as I lately

understood). Furthermore, fluorescence is the method of excellence for detecting

concentration in microdevices. The amounts of protein used in microsystems are often very

dichroic mirror

light source

pinholes

detector

objective

specimennot in focal plane

focal plane

Figure 2.2. Scheme of an epifluorescence confocal microscope (like the one mounted in this work).

The light to and from the specimen passes by an objective. A dichroic mirror passes the incident light

and reflects the fluorescence.

chapter 3. Adsorption in Microsystems: the Effect of the Surface to Volume Ratio66

remark), since a moderate depletion of the bulk concentration cannot be detected with the

confocal microscope. To obtain max and K (necessary for the simulations fitting the

experimental kinetics of adsorption shown in Fig. 3.7c and carried out with the geometry of

Fig. 3.1b), the adsorption isotherm equation (eq. (3-3)) is linearised as follows:34

!"

!!" #

!

!" #$% #$%

& (3-15)

1000x10-12

800

600

400

200

0

eq/m

ol m

-2

7x10-36543210

C°/mol·m-3

(a)

8x106

7

6

5

4

3

2

1

0

C°/

eq/m

-1

7x10-36543210

C°/mol·m-3

(b)

10x10-10

8

6

4

2

0

/mol

m-2

302520151050

time/min

(c)

! C°/mol·m-3 = 10-3

C°/mol·m-3 = 6.7 $%10-4

C°/mol·m-3 = 6.7 $%10-5

C°/mol·m-3 = 6.7 $%10-6

Figure 3.7. (a) Isotherm of adsorption of anti-

rabbit IgG on laser-ablated PET obtained from

the experimental results at 30 min of Fig. 3.7c.

The fit has the only purpose of illustrating the

trend of the isotherm. (b) (inside) Linearisation

of the adsorption isotherm, following eq. (15):

regression coefficient = 0.999, slope = 1.08 $ 109

m2 mol-1, intercept = 9.40 $ 104 m-1, from which

max = 9.26 $ 10-10 mol·m-2 and K = 1.15 $ 104

m3·mol-1. (c) Simulations (lines) compared with

experimental results (markers). Calculations are

run with D = 4 $ 10-11 m2 sec-1, max = 9.26 $ 10-

10 mol m-2, K = 1.15 $ 104 m3 mol-1 (kon = 11.5

m3 mol-1 sec-1 and koff = 10-3 sec-1). The initial

concentrations for experiments and simulations

are C° = 10-3, 6.67 $ 10-4, 6.67 $ 10-5, and 6.67 $

10-6 mol·m-3.


Conclusions

In systems where we have a large surface/volume ratio, the kinetics of adsorption is much faster.

So far, we did not consider mass transfer.

We shall now consider diffusion to the surface


Schematic of a sensor immersed in analyte solution.The diffusion of analyte particles towards a planar device is 1D, towards a cylindrical nanowire is 2D, and that to a spherical surface nanosphere is 3D. Equilibrium analyte concentration is assumed at a distance W from the sensor surface.

Sensor geometry

APPLIED PHYSICS LETTERS 88, 233120 (2006)


Diffusion - Adsorption

Random walkdiffusion

c(x,t)t

= D 2c(x,t)x2

(t) t

= D c(x,t) x

��

��x=0

= �J(t)

Differential equations


csurf

Rate of adsorption larger than the arrival rate

Adsorption-diffusion

Surface layer(nm)

z

Adsorbed layer

cbulk

Diffusion layerμm


Adsorption -diffusion

Steady-state : Surface accumulation = Adsorption flux = Diffusion flux

d(t)dt

= koncsurfmax = J =

D

cbulk csurf( )

<< eq

koncbulk >> konc

bulk + koff( ) << / ta

Approximation at short times

<< eq

We shall neglect the desorption

csurf =Dcbulk

konmax + D


Adsorption -diffusion

Steady-state

<< eq

d(t)dt

=konDc

bulkmaxkonmax + D

Da << 1 (t) = eqttd

Diffusion faster than adsorption

Da >> 1 (t) =Dcbulk

tDiffusion control

Da : Damköhler number!(t) = koncbulk!max

1+ kon!max!D

"

#

$$$

%

&

'''t =

!eq1+ Da"#$

%&'ttd


Perturbation approach

= 2Dt

We take into account the growth of the diffusion layer in unstirred solutions.

<< eq

Approximate time -dependence

!(t) = koncbulk!max

1+ kon!max2tD

"

#

$$$

%

&

'''t = !eq

ttd

"#$

%&'

1

1+ kon!max2tD


Influence of the diffusion

(t) =

eq1+ Da

�

��

�

��ttd

Steady state :

(t) =konc

bulkmax

1+ konmax2tD

�

�

��

�

�

��

tUnstirred :

Ideal stirring :

K=109 M–1

kon=106 M–1·s–1

cbulk = 0.1 pM

D=10–10 m2·s–1

δ = 100 μm

Da = 1

max 109 mol·m2

<< eq

= eq 1 exp t / [ ]

!


Surface equilibrium - Surface concentration depletion

Diffusion

Surface layer(nm)

z

Adsorbed layer

cbulk

Diffusion layerμm

csurf


Steady state - Surface equilibrium

z << Kmax

Flux = Increase of both the adsorbed and the surface layer


Surface concentration

d!(t)dt

+ z dcsurf (t)dt

= " J = D!

cbulk " csurf( )

!(t) = K!maxcbulk 1" exp " Dt

! z + K!max( )#

$%

&

'(

#

$%%

&

'((

!(t) = !eq 1" exp " 1Da

ttd

#$%

&'(

)

*+

,

-.

)

*++

,

-..

d!(t)dt

1+ zK!max

"#$

%&'+ D

!K!max

"#$

%&'!(t) = D

!cbulk


Low surface coverageSemi-infinite diffusion

independent of bulk concentration

0.5

0.4

0.3

0.2

0.1

0.0

/ eq

6543210

time/sec

simulation analytical solution

(a)

P. Delahay et al., J. Am. Chem. Soc., 79 (1957) 2355

K = 2.5 109 M1

cbulk = 4pM

�

D = 5 1010 m2 s1

max = 3.5 1011 mol m2

kon = 2.5 108 m3 mol1 s1

koff =100 s1

<< 1

LabChip, 2005, 5, 1096–1103

!(t) = !eq 1" exp(a2Dt)erfc(a Dt )#$ %&

a = 1 /


Error function

erf(x) 2

et2dt

0

x

The error function is encountered in integrating the normal distribution, i.e. the normalised form of the Gaussian function.

1.0

0.5

0.0

-0.5

-1.0

erf(x)

420-2-4x


General case - Simulation1.0

0.8

0.6

0.4

0.2

0.0

/ eq

2.01.51.00.50.0

1/2

= 10 = 1 = 0.1

�

= Kc bulk

�

=4DtK 2max

2

Simulation vs Reinmuth, W. H. J. Phys. Chem., 1961, 65, 473LabChip, 2005, 5, 1096–1103

Dimensionless time


Stirring << 1

K=109 M–1

kon=106 M–1·s–1

cbulk = 0.1 pM

D=10–10 m2·s–1

δ = 100 μm

Da = 1

max 109 mol·m2

Ideal stirring :

Stirred :

Unstirred fast kinetics :

Unstirred/perturbation :

!(t) = !eq 1" exp "t / td( )#$ %&

!(t) = !eq 1" exp " 1Da

ttd

#$%

&'(

)

*+

,

-.

)

*++

,

-..

!(t) = !eq 1" exp(a2Dt)erfc(a Dt )#$ %&

1.0

0.8

0.6

0.4

0.2

0.0

/

eq

1000080006000400020000

t /s

Kinetic-Steady state diffusion Diffusion control Kinetic-Time dependent

diffusion layer thickness

!(t) = !eq 1" exp " D / 2tK!max

#$%

&'(t

)

*++

,

-..

)

*++

,

-..


Diffusion vs kinetics

160x10-15

140

120

100

80

60

40

20

0

/mol·m-2

6543210time/sec

a

b

ec d

�

K = 2.5 106 M1

kon / M1s1 koff / s

−1

a 2.5 105 0.1b 2.5 106 1c 2.5 107 10d 2.5 108 100e 2.5 109 1000

When the desorption time increases, it becomes kinetically controlled

D = 5 1010 m2 s1

max = 3.5 1011 mol m2

K = 2.5 109 M1

cbulk = 4pM

<< 1

LabChip, 2005, 5, 1096–1103


General case

d(t)dt

= koncsurfmax 1( ) koffmax = J =

D

cbulk csurf( )

Steady state equation

d(t)dt

= koncsurf 1( ) koff =

Dmax

cbulk csurf( )

Surface concentration

csurf =cbulk + Da / K( )1+ Da 1( )


General case


d(t)dt

=konc

bulk + Dakoff ( ) 1( )1+ Da 1( )

koff =konc

bulk koncbulk + koff( )

1+ Da 1( )

1+ Da 1!!( )! !" ! +1( )"

#$

%

&'d! = dt

td

Da! ! 1+ Da" +1

"#$

%&'ln 1!! " +1

""#$

%&'

"#$

%&'

=t " +1( )td

Explicit solution


General case

Da = 1td =1000sFull line = Ideal stirringDotted line = Explicit

Explicit solution

1.0

0.8

0.6

0.4

0.2

0.0

/

eq

300025002000150010005000

t /s

=0.1 =1 =10

Da! ! 1+ Da" +1

"#$

%&'ln 1!! " +1

""#$

%&'

"#$

%&'

=t " +1( )td


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