text:
# 40 CFR § 86.1868-12 - CO2 credits for improving the efficiency of air conditioning systems.

§ 86.1868-12 CO2 credits for improving the efficiency of air conditioning systems.

Manufacturers may generate credits applicable to the CO2 fleet average program described in § 86.1865-12 by implementing specific air conditioning system technologies designed to reduce air conditioning-related CO2 emissions over the useful life of their passenger automobiles and/or light trucks (including MDPV). The provisions of this section do not apply for non-MDPV heavy-duty vehicles. Credits shall be calculated according to this section for each air conditioning system that the manufacturer is using to generate CO2 credits. Manufacturers may also generate early air conditioning efficiency credits under this section for the 2009 through 2011 model years according to the provisions of § 86.1871-12(b). For model years 2012 and 2013 the manufacturer may determine air conditioning efficiency credits using the requirements in paragraphs (a) through (d) of this section. For model years 2014 through 2016 the eligibility requirements specified in either paragraph (e) or (f) of this section must be met before an air conditioning system is allowed to generate credits. For model years 2017 through 2019 the eligibility requirements specified in paragraph (f) of this section must be met before an air conditioning system is allowed to generate credits. For model years 2020 and later the eligibility requirements specified in paragraph (g) of this section must be met before an air conditioning system is allowed to generate credits.

(a)

(1) 2012 through 2016 model year air conditioning efficiency credits are available for the following technologies in the gram per mile amounts indicated in the following table:

Air conditioning technology Credit
value
(g/mi)
Reduced reheat, with externally-controlled, variable-displacement compressor (e.g. a compressor that controls displacement based on temperature setpoint and/or cooling demand of the air conditioning system control settings inside the passenger compartment). 1.7
Reduced reheat, with externally-controlled, fixed-displacement or pneumatic variable displacement compressor (e.g. a compressor that controls displacement based on conditions within, or internal to, the air conditioning system, such as head pressure, suction pressure, or evaporator outlet temperature). 1.1
Default to recirculated air with closed-loop control of the air supply (sensor feedback to control interior air quality) whenever the ambient temperature is 75 °F or higher: Air conditioning systems that operated with closed-loop control of the air supply at different temperatures may receive credits by submitting an engineering analysis to the Administrator for approval. 1.7
Default to recirculated air with open-loop control air supply (no sensor feedback) whenever the ambient temperature is 75 °F or higher. Air conditioning systems that operate with open-loop control of the air supply at different temperatures may receive credits by submitting an engineering analysis to the Administrator for approval. 1.1
Blower motor controls which limit wasted electrical energy (e.g. pulse width modulated power controller). 0.9
Internal heat exchanger (e.g. a device that transfers heat from the high-pressure, liquid-phase refrigerant entering the evaporator to the low-pressure, gas-phase refrigerant exiting the evaporator). 1.1
Improved condensers and/or evaporators with system analysis on the component(s) indicating a coefficient of performance improvement for the system of greater than 10% when compared to previous industry standard designs). 1.1
Oil separator. The manufacturer must submit an engineering analysis demonstrating the increased improvement of the system relative to the baseline design, where the baseline component for comparison is the version which a manufacturer most recently had in production on the same vehicle design or in a similar or related vehicle model. The characteristics of the baseline component shall be compared to the new component to demonstrate the improvement. 0.6

(2) 2017 and later model year air conditioning efficiency credits are available for the following technologies in the gram per mile amounts indicated for each vehicle category in the following table:

Air conditioning technology Passenger automo-
biles
(g/mi)
Light
trucks
(g/mi)
Reduced reheat, with externally-controlled, variable-displacement compressor (e.g. a compressor that controls displacement based on temperature setpoint and/or cooling demand of the air conditioning system control settings inside the passenger compartment). 1.5 2.2
Reduced reheat, with externally-controlled, fixed-displacement or pneumatic variable displacement compressor (e.g. a compressor that controls displacement based on conditions within, or internal to, the air conditioning system, such as head pressure, suction pressure, or evaporator outlet temperature). 1.0 1.4
Default to recirculated air with closed-loop control of the air supply (sensor feedback to control interior air quality) whenever the ambient temperature is 75 °F or higher: Air conditioning systems that operated with closed-loop control of the air supply at different temperatures may receive credits by submitting an engineering analysis to the Administrator for approval. 1.5 2.2
Default to recirculated air with open-loop control air supply (no sensor feedback) whenever the ambient temperature is 75 °F or higher. Air conditioning systems that operate with open-loop control of the air supply at different temperatures may receive credits by submitting an engineering analysis to the Administrator for approval. 1.0 1.4
Blower motor controls which limit wasted electrical energy (e.g. pulse width modulated power controller). 0.8 1.1
Internal heat exchanger (e.g. a device that transfers heat from the high-pressure, liquid-phase refrigerant entering the evaporator to the low-pressure, gas-phase refrigerant exiting the evaporator). 1.0 1.4
Improved condensers and/or evaporators with system analysis on the component(s) indicating a coefficient of performance improvement for the system of greater than 10% when compared to previous industry standard designs). 1.0 1.4
Oil separator. The manufacturer must submit an engineering analysis demonstrating the increased improvement of the system relative to the baseline design, where the baseline component for comparison is the version which a manufacturer most recently had in production on the same vehicle design or in a similar or related vehicle model. The characteristics of the baseline component shall be compared to the new component to demonstrate the improvement. 0.5 0.7
Advanced technology air conditioning compressor with improved efficiency relative to fixed-displacement compressors achieved through the addition of a variable crankcase suction valve. 1.1 1.1

(b) Air conditioning efficiency credits are determined on an air conditioning system basis. For each air conditioning system that is eligible for a credit based on the use of one or more of the items listed in paragraph (a) of this section, the total credit value is the sum of the gram per mile values for the appropriate model year listed in paragraph (a) of this section for each item that applies to the air conditioning system.

(1) In the 2012 through 2016 model years the total credit value for an air conditioning system for passenger automobiles or light trucks may not be greater than 5.7 grams per mile.

(2) In the 2017 and later model years the total credit value for an air conditioning system may not be greater than 5.0 grams per mile for any passenger automobile or 7.2 grams per mile for any light truck.

(c) The total efficiency credits generated by an air conditioning system shall be calculated separately for passenger automobiles and light trucks according to the following formula:

Total Credits (Megagrams) = (Credit × Production × VLM) ÷ 1,000,000
Where:
Credit = the CO2 efficiency credit value in grams per mile determined in paragraph (b) or (e) of this section, whichever is applicable.
Production = The total number of passenger automobiles or light trucks, whichever is applicable, produced with the air conditioning system to which to the efficiency credit value from paragraph (b) of this section applies.
VLM = vehicle lifetime miles, which for passenger automobiles shall be 195,264 and for light trucks shall be 225,865.

(d) The results of paragraph (c) of this section, rounded to the nearest whole number, shall be included in the manufacturer's credit/debit totals calculated in § 86.1865-12(k)(5).

(e) For the 2014 through 2016 model years, manufacturers must validate air conditioning credits by using the Air Conditioning Idle Test Procedure according to the provisions of this paragraph (e) or, alternatively, by using the AC17 reporting requirements specified in paragraph (f) of this section. The Air Conditioning Idle Test Procedure is not applicable after the 2016 model year.

(1) For each air conditioning system selected by the manufacturer to generate air conditioning efficiency credits, the manufacturer shall perform the Air Conditioning Idle Test Procedure specified in § 86.165-12 of this part.

(2) Using good engineering judgment, the manufacturer must select the vehicle configuration to be tested that is expected to result in the greatest increased CO2 emissions as a result of the operation of the air conditioning system for which efficiency credits are being sought. If the air conditioning system is being installed in passenger automobiles and light trucks, a separate determination of the quantity of credits for passenger automobiles and light trucks must be made, but only one test vehicle is required to represent the air conditioning system, provided it represents the worst-case impact of the system on CO2 emissions.

(3) The manufacturer shall determine an idle test threshold (ITT) for the tested vehicle configuration. A comparison of this threshold value with the CO2 emissions increase recorded over the Air Conditioning Idle Test Procedure in § 86.165-12 determines the total credits that may be generated by an air conditioning system. The manufacturer may choose one of the following idle test threshold (ITT) values for an air conditioning system:

(i) 14.9 grams per minute; or

(ii) The value determined from the following equation, rounded to the nearest tenth of a gram per minute:

$\text{Idle Test Threshold}\left(\mathrm{ITT}\right)=20.5-\left(1.58×\text{Displacement}\right)$

Where:
Displacement = the engine displacement of the test vehicle, expressed in liters and rounded to the nearest one tenth of a liter.

(4)

(i) If the CO2 emissions value determined from the Idle Test Procedure in § 86.165-12 is less than or equal to the idle test threshold (ITT) determined in paragraph (e)(3) of this section, the total CO2 efficiency credit value (Credit) for use in paragraph (c) of this section shall be the applicable value determined in paragraph (b) of this section.

(ii) If the CO2 emissions value determined from the Idle Test Procedure in § 86.165-12 is greater than the idle test threshold (ITT) determined in paragraph (e)(3) of this section, the total CO2 efficiency credit value (Credit) for use in paragraph (c) of this section shall be determined using the following formula:

$\text{Credit}=\mathrm{TC}V×\left[1-\left(\frac{ITP-ITT}{6.4}\right)\right]$

Where:
Credit = The CO2 efficiency credit value (Credit) that must be used in paragraph (c) of this section to calculate the total credits (in Megagrams) of air conditioning efficiency credits;
TCV = The total CO2 efficiency credit value determined according to paragraph (b) of this section; and
ITP = the increased CO2 emissions determined from the Idle Test Procedure in § 86.165-14.
ITT = the idle test threshold determined in paragraph (e)(3) of this section and rounded to the nearest one tenth of a gram per minute:

(iii) Air conditioning systems that record an increased CO2 emissions value on the Idle Test Procedure in § 86.165-14 that is greater than or equal to the idle test threshold (ITT) determined in paragraph (e)(3) of this section plus 6.4 grams per minute are not eligible for an air conditioning efficiency credit.

(5) Air conditioning systems with compressors that are solely powered by electricity shall submit Air Conditioning Idle Test Procedure data to be eligible to generate credits in the 2014 and later model years, but such systems are not required to meet a specific threshold to be eligible to generate such credits, as long as the engine remains off for a period of at least 2 cumulative minutes during the air conditioning on portion of the Idle Test Procedure in § 86.165-12(d).

(f) AC17 reporting requirements. Manufacturers may use the provisions of this paragraph (f) as an alternative to the use of the Air Conditioning Idle Test to demonstrate eligibility to generate air conditioning efficiency credits for the 2014 through 2016 model years. This paragraph (f) is required for the 2017 through 2019 model years.

(1) The manufacturer shall perform the AC17 test specified in 40 CFR 1066.845 on each unique air conditioning system design and vehicle platform combination (as those terms are defined in § 86.1803) for which the manufacturer intends to accrue air conditioning efficiency credits. The manufacturer must test at least one unique air conditioning system within each vehicle platform in a model year, unless all unique air conditioning systems within a vehicle platform have been previously tested. A unique air conditioning system design is a system with unique or substantially different component designs or types and/or system control strategies (e.g., fixed displacement vs. variable displacement compressors, orifice tube vs. thermostatic expansion valve, single vs. dual evaporator, etc.). In the first year of such testing, the tested vehicle configuration shall be the highest production vehicle configuration within each platform. In subsequent model years the manufacturer must test other unique air conditioning systems within the vehicle platform, proceeding from the highest production untested system until all unique air conditioning systems within the platform have been tested, or until the vehicle platform experiences a major redesign. Whenever a new unique air conditioning system is tested, the highest production configuration using that system shall be the vehicle selected for testing. Air conditioning system designs which have similar cooling capacity, component types, and control strategies, yet differ in terms of compressor pulley ratios or condenser or evaporator surface areas will not be considered to be unique system designs. The test results from one unique system design may represent all variants of that design. Manufacturers must use good engineering judgment to identify the unique air conditioning system designs which will require AC17 testing in subsequent model years. Results must be reported separately for all four phases (two phases with air conditioning off and two phases with air conditioning on) of the test to the Environmental Protection Agency, and the results of the calculations required in 40 CFR 1066.845 must also be reported. In each subsequent model year additional air conditioning system designs, if such systems exist, within a vehicle platform that is generating air conditioning credits must be tested using the AC17 procedure. When all unique air conditioning system designs within a platform have been tested, no additional testing is required within that platform, and credits may be carried over to subsequent model years until there is a significant change in the platform design, at which point a new sequence of testing must be initiated. No more than one vehicle from each credit-generating platform is required to be tested in each model year.

(2) The manufacturer shall also report the following information for each vehicle tested: the vehicle class, model type, curb weight, engine displacement, transmission class and configuration, interior volume, climate control system type and characteristics, refrigerant used, compressor type, and evaporator/condenser characteristics.

(g) AC17 validation testing and reporting requirements. For 2020 and later model years, manufacturers must validate air conditioning credits by using the AC17 Test Procedure in 40 CFR 1066.845 as follows:

(1) For each air conditioning system (as defined in § 86.1803) selected by the manufacturer to generate air conditioning efficiency credits, the manufacturer shall perform the AC17 Air Conditioning Efficiency Test Procedure specified in 40 CFR 1066.845, according to the requirements of this paragraph (g).

(2) Complete the following testing and calculations:

(i) Perform the AC17 test on a vehicle that incorporates the air conditioning system with the credit-generating technologies.

(ii) Perform the AC17 test on a vehicle which does not incorporate the credit-generating technologies. The tested vehicle must be similar to the vehicle tested under paragraph (g)(2)(i) of this section and selected using good engineering judgment. The tested vehicle may be from an earlier design generation. If the manufacturer cannot identify an appropriate vehicle to test under this paragraph (g)(2)(ii), they may submit an engineering analysis that describes why an appropriate vehicle is not available or not appropriate, and includes data and information supporting specific credit values, using good engineering judgment.

(iii) Subtract the CO2 emissions determined from testing under paragraph (g)(1)(i) of this section from the CO2 emissions determined from testing under paragraph (g)(1)(ii) of this section and round to the nearest 0.1 grams/mile. If the result is less than or equal to zero, the air conditioning system is not eligible to generate credits. If the result is greater than or equal to the total of the gram per mile credits determined in paragraph (b) of this section, then the air conditioning system is eligible to generate the maximum allowable value determined in paragraph (b) of this section. If the result is greater than zero but less than the total of the gram per mile credits determined in paragraph (b) of this section, then the air conditioning system is eligible to generate credits in the amount determined by subtracting the CO2 emissions determined from testing under paragraph (g)(1)(i) of this section from the CO2 emissions determined from testing under paragraph (g)(1)(ii) of this section and rounding to the nearest 0.1 grams/mile.

(3) For the first model year for which an air conditioning system is expected to generate credits, the manufacturer must select for testing the projected highest-selling configuration within each combination of vehicle platform and air conditioning system (as those terms are defined in § 86.1803). The manufacturer must test at least one unique air conditioning system within each vehicle platform in a model year, unless all unique air conditioning systems within a vehicle platform have been previously tested. A unique air conditioning system design is a system with unique or substantially different component designs or types and/or system control strategies (e.g., fixed-displacement vs. variable displacement compressors, orifice tube vs. thermostatic expansion valve, single vs. dual evaporator, etc.). In the first year of such testing, the tested vehicle configuration shall be the highest production vehicle configuration within each platform. In subsequent model years the manufacturer must test other unique air conditioning systems within the vehicle platform, proceeding from the highest production untested system until all unique air conditioning systems within the platform have been tested, or until the vehicle platform experiences a major redesign. Whenever a new unique air conditioning system is tested, the highest production configuration using that system shall be the vehicle selected for testing. Credits may continue to be generated by the air conditioning system installed in a vehicle platform provided that:

(i) The air conditioning system components and/or control strategies do not change in any way that could be expected to cause a change in its efficiency;

(ii) The vehicle platform does not change in design such that the changes could be expected to cause a change in the efficiency of the air conditioning system; and

(iii) The manufacturer continues to test at least one unique air conditioning system within each platform using the air conditioning system, in each model year, until all unique air conditioning systems within each platform have been tested.

(4) Each air conditioning system must be tested and must meet the testing criteria in order to be allowed to generate credits. Credits may continue to be generated by an air conditioning system in subsequent model years if the manufacturer continues to test at least one unique air conditioning system within each platform on an annual basis, unless all systems have been previously tested, as long as the air conditioning system and vehicle platform do not change substantially.

(5) AC17 testing requirements apply as follows for electric vehicles and plug-in hybrid electric vehicles:

(i) Manufacturers may omit AC17 testing for electric vehicles. Electric vehicles may qualify for air conditioning efficiency credits based on identified technologies, without testing. The application for certification must include a detailed description of the vehicle's air conditioning system and identify any technology items eligible for air conditioning efficiency credits. Include additional supporting information to justify the air conditioning credit for each technology.

(ii) The provisions of paragraph (g)(5)(i) of this section also apply for plug-in hybrid electric vehicles if they have an all electric range of at least 60 miles (combined city and highway) after adjustment to reflect actual in-use driving conditions (see 40 CFR 600.311(j)), and they do not rely on the engine to cool the vehicle's cabin for the ambient and driving conditions represented by the AC17 test.

(iii) If AC17 testing is required for plug-in hybrid electric vehicles, perform this testing in charge-sustaining mode.

(h) The following definitions apply to this section:

(1) Reduced reheat, with externally-controlled, variable displacement compressor means a system in which compressor displacement is controlled via an electronic signal, based on input from sensors (e.g., position or setpoint of interior temperature control, interior temperature, evaporator outlet air temperature, or refrigerant temperature) and air temperature at the outlet of the evaporator can be controlled to a level at 41 °F, or higher.

(2) Reduced reheat, with externally-controlled, fixed-displacement or pneumatic variable displacement compressor means a system in which the output of either compressor is controlled by cycling the compressor clutch off-and-on via an electronic signal, based on input from sensors (e.g., position or setpoint of interior temperature control, interior temperature, evaporator outlet air temperature, or refrigerant temperature) and air temperature at the outlet of the evaporator can be controlled to a level at 41 °F, or higher.

(3) Default to recirculated air mode means that the default position of the mechanism which controls the source of air supplied to the air conditioning system shall change from outside air to recirculated air when the operator or the automatic climate control system has engaged the air conditioning system (i.e., evaporator is removing heat), except under those conditions where dehumidification is required for visibility (i.e., defogger mode). In vehicles equipped with interior air quality sensors (e.g., humidity sensor, or carbon dioxide sensor), the controls may determine proper blend of air supply sources to maintain freshness of the cabin air and prevent fogging of windows while continuing to maximize the use of recirculated air. At any time, the vehicle operator may manually select the non-recirculated air setting during vehicle operation but the system must default to recirculated air mode on subsequent vehicle operations (i.e., next vehicle start). The climate control system may delay switching to recirculation mode until the interior air temperature is less than the outside air temperature, at which time the system must switch to recirculated air mode.

(4) Blower motor controls which limit waste energy means a method of controlling fan and blower speeds which does not use resistive elements to decrease the voltage supplied to the motor.

(5) Improved condensers and/or evaporators means that the coefficient of performance (COP) of air conditioning system using improved evaporator and condenser designs is 10 percent higher, as determined using the bench test procedures described in SAE J2765 “Procedure for Measuring System COP of a Mobile Air Conditioning System on a Test Bench,” when compared to a system using standard, or prior model year, component designs (SAE J2765 is incorporated by reference in § 86.1). The manufacturer must submit an engineering analysis demonstrating the increased improvement of the system relative to the baseline design, where the baseline component(s) for comparison is the version which a manufacturer most recently had in production on the same vehicle design or in a similar or related vehicle model. The dimensional characteristics (e.g., tube configuration/thickness/spacing, and fin density) of the baseline component(s) shall be compared to the new component(s) to demonstrate the improvement in coefficient of performance.

(6) Oil separator means a mechanism which removes at least 50 percent of the oil entrained in the oil/refrigerant mixture exiting the compressor and returns it to the compressor housing or compressor inlet, or a compressor design which does not rely on the circulation of an oil/refrigerant mixture for lubrication.

(7) Advanced technology air conditioning compressor means an air conditioning compressor with improved efficiency relative to fixed-displacement compressors. Efficiency gains are derived from improved internal valve systems that optimize the internal refrigerant flow across the range of compressor operator conditions through the addition of a variable crankcase suction valve.

[77 FR 63166, Oct. 15, 2012, as amended at 79 FR 23736, Apr. 28, 2014; 81 FR 73996, Oct. 25, 2016; 85 FR 25270, Apr. 30, 2020; 86 FR 34372, June 29, 2021]


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# Synopsis: Record LHC Haul Catches Double Meson Signal

A huge dataset recorded at the highest particle collision energy so-far observed resolves a puzzle by revealing two meson excited states.

Quantum chromodynamics (QCD) is currently the best description of how quarks interact via the strong nuclear force to form baryons (three-quark particles such as protons and neutrons) and mesons (quark-antiquark pairs). The predictions of QCD are well supported by observations at low energies but are still relatively untested for combinations of more massive quarks. Now, an analysis of data from the Large Hadron Collider (LHC) at CERN in Switzerland has identified the signature of two distinct states of a specific particle made up of massive quarks. The particle is the ${B}_{c}$ meson, comprising a bottom quark paired with a charm antiquark or an antibottom paired with a charm quark.

Mesons exist in various excitation states that are determined by the spin and orbital angular momenta of their constituent quarks. Two decades ago, researchers at Fermilab’s Tevatron observed the ${B}_{c}$ meson’s ground state, which physicists expected to be just the first of many possible states. Subsequent searches had mixed results: one detector at the LHC spotted what appeared to be a single ${B}_{c}$ excitation state, while another, operating at a similar energy, did not. Now the CMS Collaboration at CERN has confirmed the earlier positive result and has resolved the observed signal into a pair of closely spaced excitation states.

The ${B}_{c}$ meson combines a heavy quark (or antiquark) with a lighter partner, so researchers think that its properties should be between those of mesons comprising only light constituents and those of heavy-quark mesons. Detailed predictions are difficult, however, as the lopsided distribution of mass might make some approximations invalid. Characterization of the particle’s excited states means that certain QCD models can now be tested, advancing our understanding of quark dynamics.

This research is published in Physical Review Letters.

–Marric Stephens

Marric Stephens is a freelance science writer based in Bristol, UK.

More Features »

### Announcements

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## Subject Areas

Particles and Fields

Cosmology

Read More »

## Next Synopsis

Materials Science

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Pions could catalyze reactions between protons and neutrons, allowing the stable production of deuterons in high-energy ion-ion collisions. Read More »

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Driving an electron beam into a helical pattern with terahertz electromagnetic pulses allows researchers to measure the beam’s complete shape with femtosecond resolution. Read More »

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# Maximize It

Algebra Level 5

$\large xy+yz+xz - 2xyz$

If $$x,y$$ and $$z$$ are non-negative real numbers satisfying $$x+y+z=1$$, find the maximum value of the expression above.


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Inpainting using Variational Regularization

# Inpainting using Variational Regularization

This numerical tour explores the use of variational energies (Sobolev, total variation) to regularize the image inpaiting problem.

## Contents

Here we consider inpainting of damaged observation without noise.

## Installing toolboxes and setting up the path.

You need to download the following files: signal toolbox and general toolbox.

You need to unzip these toolboxes in your working directory, so that you have toolbox_signal and toolbox_general in your directory.

For Scilab user: you must replace the Matlab comment '%' by its Scilab counterpart '//'.

Recommandation: You should create a text file named for instance numericaltour.sce (in Scilab) or numericaltour.m (in Matlab) to write all the Scilab/Matlab command you want to execute. Then, simply run exec('numericaltour.sce'); (in Scilab) or numericaltour; (in Matlab) to run the commands.

Execute this line only if you are using Matlab.

getd = @(p)path(p,path); % scilab users must *not* execute this

Then you can add the toolboxes to the path.

getd('toolbox_signal/');
getd('toolbox_general/');

## Missing Pixels and Inpainting

Inpainting corresponds to filling holes in images.

First we load the image $$f_0 \in \RR^N$$ of $$N=n\times n$$ to be inpainted.

name = 'cameraman';
n = 256;
f0 = rescale( load_image(name, n) );

Display the original image.

clf;
imageplot(f0);

Ratio of removed pixels.

rho = .7;

Construct a random mask $$\Ga = \chi_{\Om}$$ so that $$\Ga_i=0$$ for removed pixels $$i \notin \Om$$, and $$\Ga_i=1$$ for kept pixels $$i \in \Om$$.

Gamma = rand(n)>rho;

We create the masking operator $$\Phi$$ which is a diagonal operator: $(\Phi f)_i = \Ga_i f_i$

Phi = @(f)f.*Gamma;

Compute the damaged observation $$y=\Phi(f_0)$$ (no noise is added).

y = Phi(f0);

Display the observations.

clf;
imageplot(y);

## Sobolev Impainting

We solve the inpainting problem by minimzing the Sobolev norm of the image under the constraint of matching the observation $f^\star = \uargmin{ \Phi(f) = y } E(f) = \norm{\nabla f}^2$ where $$\nabla$$ is a finite difference approximation of the gradient.

It can be shown that the solution to this problem is an harmonic function with prescribed boundary condition $\forall i \notin \Om, \quad (\Delta f^\star)_i=0 \qandq \forall i \in \Om, \quad f^\star_i = y_i.$

This problem requires the constrained minimization of a smooth function, it can thus be solved using a projected gradient descent $f^{(\ell+1)} = \Pi \pa{ f^{(\ell)} + \tau \Delta(f^{(\ell)}) }$ where $$\Pi$$ is the orthogonal projector on the constraint $$y=\Phi f$$ $(\Pi f)_i = \choice{ y_i \qifq i \in \Om, \\ f_i \qifq i \notin \Om, \\ }$

Pi = @(f)f.*(1-Gamma) + y.*Gamma;

Here $$\Delta = -\nabla^* \circ \nabla = \text{div} \circ \nabla$$ is the gradient of the Sobolev energy $$E$$.

Delta = @(f)div(grad(f));

For convergence, the gradient descent step size should satisfy: $\tau<\frac{2}{\norm{\Delta}}=\frac{1}{4}$

tau = .8/4;

Exercice 1: (check the solution) Perform the projected gradient descent. Record in a variable E the evolution of the Sobolev energy $$E$$.

exo1;

Display the decay of the energy $$E(f^{(\ell)})$$ with the iterations.

clf;
plot(E); axis('tight');
set_label('Iteration #', 'E');

Display the result.

clf;
imageplot(f, strcat(['Inpainted, SNR=' num2str(snr(f0,f),3) 'dB']));

## Inpainting with TV Regularization

A non-linear prior replaces the Sobolev energy by the TV norm, that tends to better reconstruct edges. Here we use a smoothed TV norm to avoid convergence issue with gradient descent algorithms.

The smoothed TV norm reads: $J_\epsilon(f) = \sum_x \sqrt{\norm{ \nabla f(x) }^2+\epsilon^2}$

We use a projected gradient descent to solve this problem $f^{(\ell+1)} = \Pi \pa{ f^{(\ell)} + \tau G_\epsilon(f^{(\ell)}) }$ where $$G_\epsilon$$ is the gradient of $$J_\epsilon$$, that is defined as $G_\epsilon(f) = -\text{div} N_\epsilon( \nabla f )$ where $$N_\epsilon$$ is the following normalization operator $N_\epsilon(u)_i = \frac{u_i}{ \sqrt{\norm{u_i}^2 + \epsilon^2} }$ that is applied to any vector field $$u=(u_i)_i \in \RR^{N \times 2}$$ for $$u_i \in \RR^2$$.

Regularization parameter $$\epsilon$$ for the TV norm

epsilon = 1e-2;

Define the normalization operator.

Amplitude = @(u)sqrt(sum(u.^2,3)+epsilon^2);
Neps = @(u)u./repmat(Amplitude(u), [1 1 2]);

The step size $$\tau$$, should satisfy $\tau<\frac{\epsilon}{4}.$

tau = .9*epsilon/4;

Define the gradient of $$J$$

G = @(f)-div(Neps(grad(f)));

Exercice 2: (check the solution) Perform the projected gradient descent. Record in a variable J the evolution of the TV energy $$J_\epsilon$$.

exo2;

Display the result.

clf;
imageplot(clamp(f), strcat(['SNR=' num2str(snr(f0,f),3) 'dB']));

Display the evolution of the TV norm $$J_\epsilon$$.

clf;
plot(J);
axis('tight');
set_label('Iteration #', 'J_\epsilon');

Inpainting can be used to remove objects in pictures.

n = 256;
f0 = rescale( sum(f0,3) );

Display it.

clf;
imageplot(f0);

Gamma = load_image('parrot-mask', n);
Gamma = double(rescale(Gamma)>.5);

Masking operator $$\Phi$$.

Phi = @(f)f.*Gamma;

Observation $$y=\Phi(f_0)$$.

y = Phi(f0);

Display it.

clf;
imageplot(y);

Exercice 3: (check the solution) Perform Sobolev inpainting on this image.

exo3;

Exercice 4: (check the solution) Try other methods to solve this inpainting problem. You can for instance have a look on the numerical on sparsity for deconvolution and inpainting.

exo4;


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<meta http-equiv="refresh" content="1; url=/nojavascript/"> Linear Inequalities ( Read ) | Algebra | CK-12 Foundation
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# Linear Inequalities

%
Progress
Practice Linear Inequalities
Progress
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One Variable Inequalities

Janet holds up a card that reads $2x + 6 = 16$ . Donna holds up a card that reads $2x + 6 > 16$ . Andrew says they are not the same but Donna argues with him. Show, using an example, that Andrew is correct.

### Guidance

One variable linear inequalities have a different form than one variable linear equations. Linear equations have the general form of $ax + b = c$ , where $a \ne 0$ . Linear inequalities can have one of four forms: $ax + b > c, ax + b < c, ax + b \ge c,$ or $ax + b \le c$ . You should notice the difference is that instead of an equals sign, there is an inequality symbol.

When you solve for a linear inequality, you follow the same rules as you would for a linear equation; however, you must remember one big rule: If you divide or multiply by a negative number while solving, you must reverse the sign of the inequality.

#### Example A

In the following table, a linear equation has been solved. Solve for the inequality using the similar steps. Are the steps the same? Is the inequality still true if you substitute 8 in for $p$ ?

Equation Inequality Is the inequality still true?
$2p + 4 = 20$ $2p + 4 < 20$ ?
$2p + 4 - 4 = 20 - 4$
$2p = 16$
$\frac{2p}{2}=\frac{16}{2}$
$p=8$

Solution:

Equation Inequality Is the inequality still true?
$2p + 4 = 20$ $2p + 4 < 20$ no
$2p + 4 - 4 = 20 - 4$ $2p+4{\color{red}-4}<20{\color{red}-4}$
$2p = 16$ $2p<16$
$\frac{2p}{2}=\frac{16}{2}$ $\frac{2p}{{\color{red}2}}<\frac{16}{{\color{red}2}}$
$p=8$ $p<8$

No, there is no difference in the steps used to find the two solutions.

#### Example B

In the following table, a linear equation has been solved. Solve for the inequality using the similar steps. Are the steps the same? Is the inequality still true if you substitute 6 in for $x$ ?

Equation Inequality Is the inequality still true?
$3x + 5 = 23$ $3x + 5 \ge 23$ ?
$3x + 5 - 5= 23 - 5$
$3x = 18$
$\frac{3x}{3}=\frac{18}{3}$
$x=6$

Solution:

Equation Inequality Is the inequality still true?
$3x + 5 = 23$ $3x + 5 \ge 23$ yes
$3x + 5 - 5= 23 - 5$ $3x + 5 {\color{red}-5} \ge 23{\color{red}-5}$
$3x = 18$ $3x\ge 18$
$\frac{3x}{3}=\frac{18}{3}$ $\frac{3x}{{\color{red}3}}\ge \frac{18}{{\color{red}3}}$
$x=6$ $x \ge 6$

No, there is no difference in the steps used to find the two solutions.

#### Example C

In the following table, a linear equation has been solved. Solve for the inequality using the similar steps. Are the steps the same? Is the inequality still true if you substitute 3 in for $c$ ?

Equation Inequality Is the inequality still true?
$5 - 3c = -4$ $5 - 3c \le -4$ ?
$5-5-3c =-4-5$
$-3c = -9$
$\frac{-3c}{-3}=\frac{-9}{-3}$
$c=3$

Solution:

Equation Inequality Is the inequality still true?
$5 - 3c = -4$ $5 - 3c \le -4$ yes
$5-5-3c =-4-5$ $5{\color{red}-5}-3c \le -4{\color{red}-5}$
$-3c = -9$ $-3c \le -9$
$\frac{-3c}{-3}=\frac{-9}{-3}$ $\frac{-3c}{{\color{red}-3}} \ge \frac{-9}{{\color{red}-3}}$
$c=3$ $c \ge 3$

Yes, there was a difference in the steps used for the two solutions. When dividing by –3, the sign of the inequality was reversed.

#### Concept Problem Revisited

Janet holds up a card that reads $2x + 6 = 16$ . Donna holds up a card that reads $2x + 6 > 16$ . Andrew says they are not the same but Donna argues with him. Show, using an example, that Andrew is correct.

Andrew could use a real world example. For example, say Andrew held out two $5 bills and six$1 bills. Andrew holds Janet’s card and says, "Is this true?"

Now let’s try it with Donna's inequality.

This amount of money is not greater than $16; it is just equal to$16. The two mathematical statements are not the same.

### Vocabulary

Linear Inequality
Linear inequalities can have one of four forms: $ax + b > c, ax + b < c, ax + b \ge c$ , or $ax + b \le c$ .

### Guided Practice

1. In the following table, a linear equation has been solved. Solve for the inequality using similar steps, but remember if you multiply or divide by a negative number you should reverse the inequality sign. Is the inequality still true if you substitute –10 in for $a$ ?

Equation Inequality Is the inequality still true?
$4.6a + 8.2 = 2.4a - 13.8$ $4.6a + 8.2 > 2.4a - 13.8$ ?
$4.6a + 8.2+13.8 = 2.4a - 13.8 + 13.8$
$4.6a + 22 = 2.4a$
$4.6a-4.6a + 22 = 2.4a-4.6a$
$22 =-2.2a$
$\frac{22}{-2.2}=\frac{-2.2a}{-2.2}$
$a=-10$

2. In the following table, a linear equation has been solved. Solve for the inequality using similar steps, but remember if you multiply or divide by a negative number you should reverse the inequality sign. Is the inequality still true if you substitute 6 in for $w$ ?

Equation Inequality Is the inequality still true?
$3(w + 4) = 2(3 + 2w)$ $3(w + 4) < 2(3 + 2w)$ ?
$3w + 12 = 6 + 4w$
$3w + 12-12 = 6-12 + 4w$
$3w = -6 + 4w$
$3w-4w = -6 + 4w-4w$
$-w = -6$
$\frac{-w}{-1}=\frac{-6}{-1}$
$w=6$

3. In the following table, a linear equation has been solved. Solve for the inequality using similar steps, but remember if you multiply or divide by a negative number you should reverse the inequality sign. Is the inequality still true if you substitute –10 in for $h$ ?

Equation Inequality Is the inequality still true?
$\frac{1}{3}(2-h)=4$ $\frac{1}{3}(2-h) \ge 4$ ?
$\frac{1}{3}(2-h)=4 \left(\frac{3}{3}\right)$
$\frac{1}{3}(2-h)=\frac{12}{3}$
$2-h=12$
$2-2-h=12-2$
$-h=10$
$\frac{-h}{-1}=\frac{10}{-1}$
$h=-10$

1.

Equation Inequality Is the inequality still true?
$4.6a + 8.2 = 2.4a - 13.8$ $4.6a + 8.2 > 2.4a - 13.8$ no
$4.6a + 8.2+13.8 = 2.4a - 13.8 + 13.8$ $4.6a + 8.2{\color{red}+13.8} > 2.4a - 13.8 {\color{red}+13.8}$
$4.6a + 22 = 2.4a$ $4.6a + 22 > 2.4a$
$4.6a-4.6a + 22 = 2.4a-4.6a$ $4.6a{\color{red}-4.6a} + 22 > 2.4a{\color{red}-4.6a}$
$22 =-2.2a$ $22 >-2.2a$
$\frac{22}{-2.2}=\frac{-2.2a}{-2.2}$ $\frac{22}{{\color{red}-2.2}}<\frac{-2.2a}{{\color{red}-2.2}}$
$a=-10$ $a>-10$

2.

Equation Inequality Is the inequality still true?
$3(w + 4) = 2(3 + 2w)$ $3(w + 4) < 2(3 + 2w)$ no
$3w + 12 = 6 + 4w$ $3w + 12 = 6 < 4w$
$3w + 12-12 = 6-12 + 4w$ $3w + 12{\color{red}-12} < 6 {\color{red}-12}+4w$
$3w = -6 + 4w$ $3w < -6 + 4w$
$3w-4w = -6 + 4w-4w$ $3w{\color{red}-4w} < -6+4w{\color{red}-4w}$
$-w = -6$ $-w < -6$
$\frac{-w}{-1}=\frac{-6}{-1}$ $\frac{-w}{{\color{red}-1}}>\frac{-6}{{\color{red}-1}}$
$w=6$ $w>6$

3.

Equation Inequality Is the inequality still true?
$\frac{1}{3}(2-h)=4$ $\frac{1}{3}(2-h) \ge 4$ yes
$\frac{1}{3}(2-h)=4\left(\frac{3}{3}\right)$ $\frac{1}{3}(2-h)=4\left({\color{red}\frac{3}{3}}\right)$
$\frac{1}{3}(2-h)=\frac{12}{3}$ $\frac{1}{3}(2-h)\ge \frac{12}{3}$
$2-h=12$ $2-h \ge 12$
$2-2-h=12-2$ $2{\color{red}-2}-h \ge 12{\color{red}-2}$
$-h=10$ $-h \ge 10$
$\frac{-h}{-1}=\frac{10}{-1}$ $\frac{-h}{{\color{red}-1}} \le \frac{10}{{\color{red}-1}}$
$h=-10$ $h \le-10$

### Practice

In the following table, a linear equation has been solved.

Equation Inequality Is the inequality still true?
$5.2+x+3.6=4.3$ $5.2+x+3.6 \ge 4.3$ ?
$8.8+x=4.3$
$8.8-8.8+x=4.3-8.8$
$x=-4.5$
1. Solve for the inequality using similar steps.
2. Were all of the steps the same? Why or why not?
3. Is the inequality still true if you substitute –4.5 in for $x$ ?

In the following table, a linear equation has been solved.

Equation Inequality Is the inequality still true?
$\frac{n}{4}-5=-3$ $\frac{n}{4}-5 <-3$ ?
$\frac{n}{4}-5\left(\frac{4}{4}\right)=-3\left(\frac{4}{4}\right)$
$\frac{n}{4}-\frac{20}{4}=\frac{-12}{4}$
$n-20=-12$
$n-20+20=-12+20$
$n=8$
1. Solve for the inequality using similar steps.
2. Were all of the steps the same? Why or why not?
3. Is the inequality still true if you substitute 8 in for $n$ ?

In the following table, a linear equation has been solved.

Equation Inequality Is the inequality still true?
$1-z=5(3+2z)+8$ $1-z<5(3+2z)+8$ ?
$1-z=15+10z+8$
$1-z=23+10z$
$1-z+z=23+10z+z$
$1=23+11z$
$1-23=23-23+11z$
$-22=11z$
$\frac{-22}{11}=\frac{11z}{11}$
$z=-2$
1. Solve for the inequality using similar steps.
2. Were all of the steps the same? Why or why not?
3. Is the inequality still true if you substitute –2 in for $z$ ?
1. The sum of two numbers is greater than 764. If one of the numbers is 416, what could the other number be?
2. 205 less a number is greater than or equal to 112. What could that number be?
3. Five more than twice a number is less than 20. If the number is a whole number, what could the number be?
4. The product of 7 and a number is greater than 42. If the number is a whole number less than 10, what could the number be?
5. Three less than 5 times a number is less than or equal to 12. If the number is a whole number, what could the number be?
6. Double a number and add 12 and the result will be greater than 20. The number is less than 6. What is the number?


text:
## Monday, November 16, 2015

### Microsoft Launch SEAL

It is not just the HEAT project which is working on trying to find applications of Somewhat Homomorphic Encryption in practice; others around the world are also doing this. In particular Microsoft Research Labs have just announced a project called SEAL(Simple Encrypted Arithmetic Library) which is detailed in the following paper: http://research.microsoft.com/pubs/258435/ManualHE.pdf with code available here http://sealcrypto.codeplex.com/

The Microsoft work is aimed at medical information, in particular bioinfomatics algorithms related to genomic data. The SEAL project is built around the YASHE SHE scheme based on the cyclotomic ring defined by $X^n+1$, for $n$ a power of two. The associated paper explains how large integers and floating point operations can be encoded within the YASHE scheme.

## Wednesday, November 4, 2015

### Using the Rényi divergence rather than the statistical distance in lattice-based cryptography

To analyze the efficiency of the homomorphic encryption schemes at the core of the HEAT project, we studied how the Rényi divergence can be used instead of the statistical distance. The Rényi divergence is a measure of closeness of two probability distributions. In the HEAT project, we obtained that the latter divergence can often be used as an alternative to the statistical distance in security proofs for lattice-based cryptography, and therefore in homomorphic encryption schemes.

Our contributions are as follows:

and are described in the article

Shi Bai and Adeline Langlois and Tancrède Lepoint and Damien Stehlé and Ron Steinfeld

This article has been accepted for publication at ASIACRYPT 2015 and will receive the Best Paper Award.

ASIACRYPT 2015 is one of the top-tier conference of cryptography, organized by the IACR, and will take place in December 2015 in New Zealand.


text:
# CreateDocument using a specific Evaluator (Kernel)

I was trying to build a button to create a new document that contained a dynamicaly changing table. In the parent notebook i am using another kernel - not the usual "Local". The problem is that in the created document (using CreateDocument command) the kernel that uses seems to be locked at Local giving no results (or errors).

Does enybody knows how the Created Document can use the kernel of the parent notebook ?

-
You can set Evaluator option for the new Cells, is this what you need? –  Kuba Sep 14 '13 at 14:21
Yes you are right , i suppose you mean something like that : CreateDocument[dynamicexpression, StyleDefinitions -> CurrentValue[EvaluationNotebook[], StyleDefinitions], Evaluator -> "2"] ? But i have to know also the name of the parent notebook's kernel (here i put it "2") –  tchronis Sep 14 '13 at 14:25
I'm not sure why don't you know that name already? How did you set it and name? To find the name you can use Options[EvaluationNotebook[], Evaluator] –  Kuba Sep 14 '13 at 14:28
I want to exploit all kernels of my machine with the same library and i don't like the idea of maintaining 4 different versions... Thank you @Kuba –  tchronis Sep 14 '13 at 14:31


text:
Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

# A nanofluidic device for parallel single nanoparticle catalysis in solution

## Abstract

Studying single catalyst nanoparticles, during reaction, eliminates averaging effects that are an inherent limitation of ensemble experiments. It enables establishing structure–function correlations beyond averaged properties by including particle-specific descriptors such as defects, chemical heterogeneity and microstructure. Driven by these prospects, several single particle catalysis concepts have been implemented. However, they all have limitations such as low throughput, or that they require very low reactant concentrations and/or reaction rates. In response, we present a nanofluidic device for highly parallelized single nanoparticle catalysis in solution, based on fluorescence microscopy. Our device enables parallel scrutiny of tens of single nanoparticles, each isolated inside its own nanofluidic channel, and at tunable reaction conditions, ranging from the fully mass transport limited regime to the surface reaction limited regime. In a wider perspective, our concept provides a versatile platform for highly parallelized single particle catalysis in solution and constitutes a promising application area for nanofluidics.

## Introduction

Nanoparticles are used extensively in catalysis due to their specific activity and relative abundance of highly active sites1,2,3. In this respect, great progress has been made in establishing structure–activity correlations4,5,6,7,8. Nevertheless, to date most nanoparticle catalysis studies are performed on ensembles containing thousands to billions of particles. Hence, the data obtained generally describe the response of an average nanoparticle. Additionally, nanoparticles are structurally heterogeneous at the atomic level9,10,11, and two nominally identical particles from the same synthesis batch may exhibit substantially different activity or selectivity12,13. Therefore, measurements on single nanoparticles are essential to uncover particle specific properties. In response, experimental methods that enable studies of catalytic processes on individual nanoparticles are emerging3,14,15,16,17. The most prominent methods for single particle catalysis in solution are single-molecule fluorescence microscopy (SMFM)4,12,18,19, surface-enhanced raman scattering (SERS)20,21, tip-enhanced raman spectroscopy (TERS)22,23, scan electrochemical microscopy (SECM)24, synchrotron-radiation-based infrared nanospectroscopy (SINS)6 and surface plasmon spectroscopy (SPS)25. Even though these are all effective techniques, they have limitations, such as that they require very low reactant concentrations and/or reaction rates14, that they have low throughput12, or that they probe only one particle at a time.

In this paper, we present a nanofluidic device that, in combination with epifluorescence microscopy, enables simultaneous and parallel scrutiny of catalyst activity in the liquid phase on several tens of catalyst nanoparticles, each isolated in its own nanofluidic channel. These nanochannels enable laminar flow of reactants to, and product molecules from, the individual particles and retain the fluorescent molecules within the focal plane of the microscope. In this way our approach enables parallel scrutiny of individual catalyst nanoparticles under a wide range of conditions. This is in stark contrast to single-molecule fluorescence microscopy, where reactants need to be highly diluted and experiments are carried out at—from an applications point of view—unrealistically low reactant concentrations4,12,18,19. For example, the transition between the mass transport and surface reaction limited regimes of a catalyst located in a nanoconfined volume can be visualized at the single nanoparticle level and analyzed as a function of nanoparticle size. This is important for understanding catalysis, since the overall activity of the system is dictated by the interplay between transport of reactants to the particle surface and the rate-limiting elementary step of the surface reaction, which in turn may be controlled by particle size and structure26. To this end, the nanofluidic design of our device also ensures that the reaction conditions are identical for all the catalyst particles in the device, and that up- or downstream conversion effects by neighboring particles can be completely eliminated, since each nanoparticle is isolated inside its own nanochannel (Supplementary Note 1). Using this platform and the catalytic reduction of fluorescein by borohydride over Au nanoparticles27 as model reaction, we show that in contrast to other single particle approaches4,12,17,18,19, an averaged turnover frequency (ToF) of ca. 0.1 s−1 per site can be reached, which is in the range of relevant industrial applications28. Additionally, we can probe the same nanoparticle in situ at reactant concentrations ranging from a situation where the reaction rate is fully mass transport limited to the surface reaction controlled regime.

## Results

### Layout of the nanofluidic device

At the chip level, the overall design is such that in- and outlet microchannels contact an array of nanochannels, which host single Au nanoparticles with well-defined dimensions (Fig. 1a and Supplementary Fig. 1). The nanochannel array is comprised of 11 sets of 5 nanochannels (100 nm high, 250 nm wide and 350 µm long), each decorated with a set of single Au particles with nominally identical size. Three 5-channel sets are kept empty to enable simultaneous negative control experiments on the same chip.

At the heart of our nanofluidic device are the catalytic Au nanoparticles that we have grown with high precision inside their corresponding individual nanochannels (Fig. 1b) by means of nanofabrication based on electron-beam lithography (Supplementary Note 1). All particles were grown simultaneously to a thickness of 20 nm through Au evaporation and ranged in size from 64 nm to 128 nm in diameter (Fig. 1b). Furthermore, we fabricated an elongated Au patch (139 × 1138 nm) and used it as a reference, mimicking an infinite system. As apparent from the scanning electron microscopy (SEM) images in Fig. 1b, directly after Au evaporation the nanoparticles exhibit a rough surface structure and are characterized by very small grains. Since it is impossible to structurally characterize the nanoparticles in detail after the complete assembly of the nanofluidic chip due to the anodically bonded glass lid (a step that requires 550 °C and thus is expected to significantly alter particle microstructure), we carried out a detailed characterization of a set of 66 nm particles on an open surface analog. For this purpose, we exposed these particles to the same sequence of temperature treatment and chemical cleaning steps as the particles inside the nanochannels experience during chip assembly and reaction. Evidently, the obtained nanoparticles vary slightly in size (65.8 ± 1 nm in diameter) and considerably in microstructure, ranging from single-crystalline to polycrystalline with up to approximately 10 grains (Fig. 1c and Supplementary Figs. 24). These microstructure differences originate from the annealing processing step (mimicking the anodic bonding of the lid), during which the particles recrystallize and attain a more thermodynamically stable structure with predominantly low index facets. The abundance, size, and distribution of facets are, however, expected to vary extensively between the individual particles due to the significantly different number of grains (Fig. 1c and Supplementary Fig. 2)29. To indirectly confirm this particle-to-particle heterogeneity for the actual Au nanoparticles inside the sealed nanofluidic channels, we measured their optical dark-field scattering spectra, which indeed exhibit different intensities and plasmon resonance frequencies (Supplementary Figs. 5 and 6), indicative of structural heterogeneity30.

Proceeding our discussion to the catalytic reaction, Fig. 1d depicts a CCD image of a nanochannel array, where the brightness of each channel corresponds to the fluorescence emission from fluorescein flowing through the fluidic system together with sodium borohydride in milli-Q water. The numbered labels indicate the position of the Au nanoparticles in the channel and refer to the specific particle size. Evidently, in channels decorated with an Au nanoparticle, the emission intensity is reduced downstream of the particle. This observation is in good agreement with an Au-catalyzed reduction of fluorescein with NaBH4 as the reduced form of fluorescein is non-fluorescent (Fig. 1e)27. The reaction product was analyzed in detail and verified by performing a batch reaction followed by detailed characterization using nuclear magnetic resonance spectroscopy (NMR), high resolution mass spectrometry (HRMS) and infrared spectroscopy (IR) (Supplementary Note 3, Supplementary Figs. 79).

### Reactant intensity profile in a nanochannel

For further quantitative analysis of the reaction in the nanochannels, we integrated the fluorescence intensity in each nanochannel within the area highlighted by the dashed rectangle in Fig. 1d to ultimately derive the reaction rate at each single nanoparticle (Supplementary Fig. 10). First, however, it is interesting to analyze the corresponding characteristics of the fluorescence intensity profile, that is, the fluorescein concentration profile (Supplementary Fig. 12b), along a nanochannel after reactant injection. This is uniquely facilitated by the nanofluidics approach used, since it confines the fluorophores in the microscope focal plane along the entire channel throughout the experiment. Such fluorescence intensity profiles are shown as the mean of all five particles investigated for each size in Fig. 2a. Clearly, the intensity decreases gradually along the channel and the profile exhibits an inverted sigmoidal shape with the steepest slope around the position of the nanoparticle for all sizes. This shape becomes more pronounced as the incoming fluorescein concentration is reduced during the course of the experiment (Supplementary Figs. 11 and 12). Interestingly, however, the fluorescence intensity also decreases after the particle where the reaction occurs. This is surprising at first, and to understand this behavior we resort to 1D and 3D simulations (that are in quantitative agreement) of our system for two different scenarios (Fig. 2b and Supplementary Figs. 13 and 14). The first one, under the assumption that reaction at the particle position and diffusion are the only relevant events, yields a fluorescence profile that steeply decreases at the catalyst particle position, as well as shows certain depletion upstream of the particle due to diffusion. Downstream of the catalyst, the profile is constant. However, the obtained profiles for 1D and 3D are significantly different from the experimentally observed ones downstream of the nanoparticle, where they do not capture the continued fluorescence intensity decrease. This implies a change in reactant flux after the catalyst particle, which cannot be described by reaction on the particle and diffusion alone. It is therefore relevant to discuss possible reasons for this behavior. The first one is photo bleaching during readout. However, control measurements at different excitation light intensities and exposure times show that this effect is negligible (Supplementary Fig. 15) and, furthermore, any photobleaching still occurring is accounted for by normalization using the fluorescence intensity signal from the reference channels without Au catalyst particle (Supplementary Notes 46). The second possible reason is related to the fact that fluorescein binds to SiO2 surfaces31 and thus to the inner walls of the nanochannels. Accordingly, an analytical 1D model that assumes equilibrium between reactant concentration in solution and on the nanochannel walls, combined with diffusion of the molecules adsorbed to the channel walls (see Supplementary Note 7 for details), can indeed predict a fluorescence intensity distribution along the nanochannel that is in good agreement with the experimental observation, and thus suggests that such a process is taking place.

### Turnover frequency measurements on single nanoparticles

Now turning back to single-particle activity, by measuring the flow rate in the nanochannels (Supplementary Figs. 1618) and by estimating the number of surface atoms of the nanoparticles (Supplementary Fig. 4), we were able to estimate the ToFs of each individual nanoparticle over a wide range of reaction conditions (Supplementary Fig. 12a). The correspondingly obtained ToFs for the reduction of fluorescein by borohydride (50 mM), measured simultaneously for 32 single Au nanoparticles (three channels were clogged) as a function of fluorescein concentration in the reactant flow, are summarized in Fig. 3 and Supplementary Figs. 1921. We make the following key observations. Firstly, at low fluorescein concentration, the ToF is a linear function of fluorescein concentration and does not show any sign of particle-specific response. We interpret this as the reaction being entirely mass transport limited, i.e. that flow and diffusion of fluorescein to the nanoparticle surface determines the reaction rate. We also note that the slope decreases with increasing particle size since ToF scales inversely with the amount of surface atoms in the mass transport limited regime according to the relation:

$${\mathrm{ToF}} = \frac{{C \cdot V}}{N},$$
(1)

where C is fluorescein concentration, V is the volume flowing past a particle per second, and N is the number of catalytic sites on the particle (Supplementary Fig. 22). Secondly, as we further increase the fluorescein concentration, a spread in ToF between individual particles for each set of size starts to appear. We observe a maximal ToF of 0.065 s−1, and a minimal ToF of 0.025 s−1 at identical conditions for two different nanoparticles in the set of the smallest nominal particle size of 64 nm (Fig. 3). Similarly, we find the maximum ToF at different fluorescein concentrations, which, as a general trend, increases with particle size. At the same time, both the maximal ToF and the observed particle-particle spread decreases with increasing particle size, and disappears completely for the 139 × 1138 nm Au patch. This can be understood as that for the largest nanoparticles we never reach the surface reaction limited regime and that the reaction is mostly (or completely in the case of the patches) mass transport limited. In other words, our system enables a systematic visualization of the interplay between mass transport and surface reaction control imposed by nanoconfinement, as for instance is characteristic for meso- or nanoporous support materials used in industrial catalysts.

Focusing on the smaller particle size regime only (64–91 nm, Fig. 3), we notice that the ToF starts to decrease for the highest considered fluorescein concentrations. This is in line with the Langmuir-Hinshelwood mechanism32, which states that two reactants first adsorb on the catalyst before the reaction takes place. The reaction rate is then governed by the interplay between both reactant concentrations and adsorption and desorption constants, which means that when the surface concentration of fluorescein becomes larger than the surface concentration of borohydride the ToF will decrease due to poisoning of the Au surface by fluorescein. However, the ToF dependence found for our system shows a steeper decrease at higher fluorescein concentrations than predicted purely by a Langmuir–Hinshelwood mechanism (Supplementary Fig. 23). This could either indicate that the reaction is not of the Langmuir–Hinshelwood type or that the reaction step is not rate determining, which would render the approximation invalid. Regardless, it is most certainly mainly a consequence of the tiny volume of the nanochannel (8.75 µm3 for the entire channel), which accelerates the transition between the mass transport and the surface reaction limited regimes, since when most of the molecules passing the particle also react, the limited mass transport will lead to a lower local reactant concentration in solution. This drop in local concentration in turn causes the observed steeper and more immediate decrease in activity observed at the highest fluorescein concentrations for all our experiments (Fig. 3 and Supplementary Figs. 1921).

As the next analysis step, we evaluate particle-specific ToFs as a function of the local fluorescein concentration measured at the position of each nanoparticle, based on the evolution of the fluorescence intensity distribution along the nanochannel (compare Fig. 2a and Supplementary Fig. 11) at each given time during the experiment. In other words, we derive the local fluorescein concentration from the measured fluorescence intensity at the position of each nanoparticle since it is this concentration (rather than the nominal concentration) that dictates the reaction rate. When plotting our data in this way, the activity at low local fluorescein concentrations becomes nearly equal for all particle sizes (Fig. 4). This corroborates our interpretation that in this regime the reaction is mass transport limited for all particle sizes and that site-specific activity is not important. It is also in good agreement with the calculated reaction rates in this regime, assuming a completely mass transport limited system (Eq. 1, Fig. 4, Supplementary Figs. 22 and 23). It is then not until local fluorescein concentrations higher than ~2 µM that we start to transit to the surface reaction limited regime where the ToFs become more and more particle specific, again with the largest spread in particle-specific activity for the smallest particle size considered.

### Trends in particle-specific activity

Having discussed the overall evolution of measured ToFs as a function of reactant concentration, it is interesting to further analyze the surface reaction limited regime in detail because there the single particle-specific response is most apparent and our data show that some nanoparticles are more active than others. This is exemplified in Fig. 5a, in which the maximum ToF is shown for each particle on the chip in three independent experiments (note that the maximal ToF values displayed were obtained at different nominal fluorescein concentrations for the different particle sizes). Clearly, on average, catalyst activity appears to be independent of particle size, while at the same time the spread within the population increases as particle size is decreased (Fig. 5a and Supplementary Fig. 24). To understand these trends, we remind ourselves that all considered particles are in a size regime (64–128 nm) where no traditional scalable size effects are expected. Hence, the observed spread in activity for a certain size can be assigned to the different abundance of low coordination and defect sites imposed by the microstructure5,29,33,34,35. This becomes apparent from the TEM images of a large set of representative particles that exhibit widely varying structure and abundance of highly defectuous grain boundaries, containing copious low coordination sites due to lattice orientation mismatch between crystallites (Supplementary Figs. 2, 25)33,36. The observed decreased spread in activity between particles for increasing size can also be attributed to a larger degree of mass transport limitation for larger particles due to the restricted reactant supply by the nanochannel. This restriction is evident for sizes 102–128 nm in Fig. 3, but is not found across all experiments when different reaction conditions are applied (Supplementary Figs. 1921).

To even further analyze and corroborate the single particle-specific activity, we resort to a larger data set and include nine independent measurements made on two nominally identical samples (chip 1 and chip 2—Fig. 5b and Supplementary Figs. 26 and 27). The particles retain their specific response also across a multitude of measurements, as specifically highlighted by subtracting the mean ToF from the 9 independent experiments from each data point. Furthermore, to also address this in a mathematically robust way: we calculated the correlation coefficients ($$\frac{{{\mathrm{Cov}}(f,g)}}{{\sigma _f \cdot \sigma _g}}$$, where f and g are measurement series values and σi are their standard deviation) between ToFs with subtracted mean for both chips (inset Fig. 5b and Supplementary Fig. 28). For each combination of experiments executed within one chip, we find correlation coefficients of 0.62 and 0.58 (inset in Fig. 5b and Supplementary Fig. 28). The much lower correlation coefficient of 0.22 between measurements in the two different chips thus indicates that the correlations for individual particle response between measurements is not an artefact imposed by how the measurements are performed, but a true effect related to the particles themselves.

## Discussion

We have presented a nanofluidic device that enables simultaneous and parallel scrutiny of catalyst activity in the liquid phase on several tens of individual catalyst nanoparticles at practically relevant (µM) reactant concentrations. In our platform, each catalyst nanoparticle is located in its own nanofluidic channel, which enables laminar flow of reactants to, and products from, the particles. As a second key feature, the nanofluidic design, with each nanoparticle isolated inside its own nanochannel, ensures that the reaction conditions are identical for all the catalyst particles, and that up- or downstream conversion effects by neighboring particles can be eliminated. Thirdly, the nanochannels retain the fluorescent molecules within the focal plane of the microscope during the entire experiment and thus enable the parallel scrutiny of individual catalyst nanoparticles under a wide range of reaction conditions. Specifically, as we have demonstrated here, the transition between the mass transport and surface reaction limited regimes can be visualized at the single nanoparticle level and analyzed as a function of nanoparticle size. Using the catalytic reduction of fluorescein by borohydride on Au nanoparticles as the model reaction, we found that in the mass transport limited regime all nanoparticles ranging in diameter from 60 to 130 nm exhibit essentially identical ToFs increasing linearly from 0 to ca. 0.01–0.03 s−1 per site, for local reactant concentrations increasing from 0 to 4.6 µM. In contrast, in the surface reaction limited regime, we observed distinct single particle-specific activity with some nanoparticles consistently being more active than others, as statistically verified over nine different experiments using the same nanofluidic chip. These effects could be explained by the different relative abundance of low coordination and defect sites due to a widely varying grain-microstructure between the nanoparticles, identified by ex situ TEM. Furthermore, the also observed decreased spread in activity between particles for increasing nanoparticle size could be attributed to a larger degree of mass transport control for larger particles, due to the limited reactant supply by the tiny nanochannel. In comparison with the state-of-the-art single-particle catalysis methods of SMFM (which give insights in the spatial position of the active sites due to superior spatial resolution) and SERS (which offers specific chemical information), our platform enables catalyst operation at technically relevant reactant concentrations and resolves reactant concentration gradients up- and downstream of single catalyst particles. Looking forward, we predict that our platform will not only be usable in combination with traditional fluorescence microscopy readout but also with super-resolution microscopy techniques and with micro-Raman spectroscopy to enable deeper chemical insights. Furthermore, either by utilizing nanofabrication for particles in the sub-10 nm size range37,38 or by specifically placing individual colloidal nanocrystals inside nanofluidic channels, it will become possible to extend the approach down into a size or structure range directly relevant for real catalysis beyond model systems, and to other types of reactions where a fluorescent product is formed, as demonstrated in a preliminary fashion in Supplementary Note 18 (Supplementary Figs. 29 and 30) for the catalytic oxidation of amplex ultra red to resorufin over an Au nanoparticle catalyst. Furthermore, in principle, our nanofluidic reactor platform can be applied to SMFM, SERS, and/or SPS readout to achieve a wider measurable concentration range and in this way compensate for some of the drawbacks of these techniques. Hence this platform will boost the field of single nanoparticle catalysis by enabling highly parallelized scrutiny of individual nanoparticles under perfectly controlled reaction conditions and at practically relevant reactant concentrations in solution, and by facilitating systematic visualization of the interplay between mass transport and surface reaction control imposed by nanoconfinement typical for meso- or nanoporous support materials used in industrial catalysts.

## Methods

### Nanofluidic chip fabrication

Fabrication of the nanofluidic chips was carried out in cleanroom facilities of Fed. Std.209 E Class 10–100, using electron-beam lithography (JBX-9300FS/JEOL Ltd), photolithography (MA 6/Suss MicroTec), reactive-ion etching (Plasmalab 100 ICP180/Oxford Plasma Technology and STS ICP), electron-beam evaporation (PVD 225/Lesker), magnetron sputtering (MS150/FHR), deep reactive-ion etching (STS ICP/STS) and wet oxidation (wet oxidation/Centrotherm), fusion bonding (AWF 12/65/Lenton), and dicing (DAD3350/Disco). In particular, the fabrication comprised several processing steps (Supplementary Note 1) of a 4”-silicon (p-type) wafer.

### Fluorescein reduction by borohydride on Au nanoparticles

Reagents (Fluorescein sodium salt, BioReagent, suitable for fluorescence, 46960, Sodium borohydride, powder, ≥98.0 %, 452882 and Au nanoparticles, 100 nm diameter, OD 1, stabilized suspension in citrate buffer, 742031) were purchased from Sigma Aldrich. Fresh stock solution of 2 mM fluorescein was prepared daily by dissolving 7.52 mg fluorescein salt in 10 ml of milli-Q water. Freshly prepared 100 mM aqueous solution of borohydride (37.83 mg in 10 ml) was used for each experiment due to the unstable nature of borohydride in water39. The stock solutions were then diluted and mixed to obtain a reaction mixture containing concentrations of fluorescein varying from 2.3 to 7.8 µM and 50 mM borohydride. Further, the reaction mixtures were directly injected into one of the inlets of the nanofluidic chip. The fluorescein concentration was confirmed by absorption measurements using a nanodrop 1000 spectrophotometer.

Before starting each experiment, the nanofluidic chip was flushed for 20 min with a solution of 4.3 wt% ammonia and 4.3 wt% hydrogen peroxide in milli-Q water to clean the nanoparticle surfaces. The chip was then flushed with milli-Q water twice to remove residues of the cleaning reagents. Measurements were then performed by flowing the reaction solution through an array of nanochannels, each containing a single Au nanoparticle, and evaluating the fluorescence signal after each particle. The evaluation was performed using in-house Matlab scripts to calculate ToFs for each channel/particle. The flow rate was during measurements kept constant at 145 µm per s (Supplementary Fig. 17).

### Characterization of Au nanoparticles

The microstructure of the used Au nanoparticles was characterized using a TEM, FEI Tecnai T20 operated at the accelerating voltage 200 kV. Nanoparticle sizes were determined using an SEM, Zeiss Supra 55—EDX operated at the accelerating voltage 15 kV. Plasmonic spectra of the Au nanoparticles were measured with dark-field scattering, using a Nikon eclipse LV150N microscope with a Nikon TU plan ELWD 50x NA = 0.6 objective, a 50 W halogen lamp, an Andor Shamrock SR-193i spectrometer with grating blaze 800 nm and a grating density of 150 g per mm and an Andor Newton 920 CCD camera. Additional details are found in Supplementary Note 2.

### Synthesis, purification and characterization of reduced fluorescein

An aqueous solution of fluorescein (50 mM) was stirred at room temperature with sodium borohydride (1 M) and Au nanoparticles (100 nm, 9.5 × 107 particles per ml) for 3 h, protected from light, to completely reduce the fluorescein (Supplementary Fig. 7). The reaction mixture was then acidified to pH 4 by addition of 10% citric acid solution and extracted three times with 10 mL ethyl acetate. Organic solvent, as well as citric acid solution, was degassed with nitrogen prior the work up of the product to suppress aerobic oxidation. The organic aliquots were combined, dried over anhydrous sodium sulphate and concentrated in vacuo. The residue was dried under high vacuum for 24 hours to yield the air sensitive product as a yellow film. The product residue was characterized with NMR, HRMS, and IR (Supplementary Note 3).

### Diffusion and reaction simulations

1D and 3D simulations of diffusion and the reaction in a nanochannel were performed using in-house Matlab scripts (1D) and computational fluid dynamics (CFD) code ANSYS Fluent 15.0.7 (3D). Details are found in Supplementary Notes 7, 8, and 10.

## Data availability

The data that support the findings of this study are available from the corresponding authors upon request.

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## Acknowledgements

We acknowledge financial support from the Knut and Alice Wallenberg Foundation via project 2015.0055. This research (C.L.) has also received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (678941/SINCAT). We thank the Knut and Alice Wallenberg Foundation for their support of the infrastructure in the MC2 nanofabrication laboratory at Chalmers. We also thank Vladimir P. Zhdanov for fruitful discussions. Part of this work was performed at MC2 cleanroom facility at Chalmers and at the Chalmers Materials Analysis Laboratory, CMAL. Open access funding provided by Chalmers University of Technology.

## Author information

Authors

### Contributions

S.L designed, performed and evaluated experiments including catalysis measurements and simulations in 1D, and wrote the paper. J.F. fabricated nanofluidic chips and characterized nanoparticles with SEM. S.N. characterized nanoparticles with TEM and dark-field scattering. A.R. characterized the reaction product with MS, NMR and IR spectroscopy. B.D. assisted in finding suitable fluorescent probes. H.St. performed flow and reaction simulations in 3D. H.Su. supervised the project and wrote the paper. C.L. supervised the project and wrote the paper. F.W. supervised the project and wrote the paper. All authors discussed the results and contributed to the paper.

### Corresponding authors

Correspondence to Christoph Langhammer or Fredrik Westerlund.

## Ethics declarations

### Competing interests

The authors declare no competing interests.

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Levin, S., Fritzsche, J., Nilsson, S. et al. A nanofluidic device for parallel single nanoparticle catalysis in solution. Nat Commun 10, 4426 (2019). https://doi.org/10.1038/s41467-019-12458-1

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