# Presentation: 318171

## STATISTICAL ANALYSIS AND PARAMETERIZATION OF NUCLEATION RATE MEASUREMENTS: INTERPRETATION AND CORRELATION WITH VAPOR COMPOSITION AND TEMPERATURE

***Robert McGraw***
***Atmospheric Sciences Division***
***Brookhaven National Laboratory, Upton NY 11973***

- 1.  Isothermal nucleation theorem and selection of vapor composition
-      coordinates
- 2.  Principal Components Analysis (PCA): Illustrated use for the
-      (p-toluic acid /sulfuric acid /water) ternary measurements of Zhang et al.
- 3.  Non-isothermal nucleation theorem and selection of the temperature
-      coordinate.  Illustrated for water vapor measurements of Wölk and Strey.
- 4.  Key question:  Can these new methods be extended beyond analysis of
-      laboratory nucleation measurements to field measurements of atmospheric
-      new particle formation?

## const T

_*const T*_

*nucleation rate*

*number conc.*
*(vapor)*

*saturation ratio*

***Example:****** water vapor measurements of ***
***W***ö***lk and Strey, ***
*** ******J. Phys. Chem. ***_***B105***_***, 11683 (2001).***

*number of molecules*
*in critical nucleus*

***T=250K***

## const T, n1(j)

*const T, n**1**(j)*

*nucleation rate*

*monomer number conc.*
*species “i ” in vapor*

*saturation ratio*
*species “i ” *

*number of molecules of*
*species “i ”  present in the critical nucleus*

*small kinetic term (between 0 and 1)*
*related to the direction of nucleation flux *
*over the cluster energy surface*

- CORRELATING NUCLEATION RATE WITH VAPOR COMPOSITION

## ILLUSTRATION FOR A BINARY NUCLEATION RATE

**ILLUSTRATION FOR A BINARY NUCLEATION RATE**

- Flux through critical saddle point region illustrated
- for binary nucleation.  From McGraw and Wu 2003.

## log[J ] has strong propensity to be multi-linear 

**INTEGRATED FORM OF THE KINETIC NUCLEATION THEOREM **
**FOR MULTICOMPONENT NUCLEATION**

*1 component (e.g., water)*

*2 components (e.g., sulfuric acid/ water)*

*3 components (e.g., sulfuric acid/ organic acid/ water)*

*log[J ] has strong propensity to be multi-linear *
*in coordinates:*
*log[concentration species(i)]*

## EXAMPLE : TERNARY NUCLEATION INVOLVING AN ORGANIC ACID 

**EXAMPLE :**** TERNARY NUCLEATION INVOLVING AN ORGANIC ACID **

*CRITICAL NUCLEUS *
*MOLECULAR CONTENT*
*(from slopes or relative sensitivities: ∂ Log J / ∂ Log S**i** )*

- # H2SO4 = 8 molecules
- # p-toluic acid = 1 molecule!

**Data of Zhang et al., *****Science ***_**304**_**, 1487 (2004)**

- NT motivated replot on log-log scale

- H2SO4 (*10**9** molecule cm**-3*)

- (*a* + ***a* = 8.3)
- (*c* + ***c* = 1.7)

## N-theorem coordinates

*x* = Log10[*H**2**SO**4**, molecules cm**-3*]
*y* = Log10[*Organic, ppb*]
*z* = Log10[*J, cm**-3*]

- Measurements (flexible data collection):
- {*x**i**, y**i**, z**i*}   *i* = *1*,*N*

- centered coordinate vectors:
- w(i)= {*x**i**- ****x**, y**i**- ****y**, z**i**- ****z*}   *i* = *1*, *N*
- {***x**, ****y**, ****z** *} is centroid

***COVARIANCE MATRIX***:

***EIGENVALUES***
***(sorted principal values)***
*********1  ******>  ************2 ******> ************3 ***
***EIGENVECTORS***
***(principal components)***
**      ****{ v****1****, v****2****, v****2****}**

*N-theorem coordinates*

- i = w(i) • v1 , etc.

*principal coordinates*

## COMPARING MEASUREMENTS IN THE PRINCIPAL FRAME AND IN N-THEOREM COORDINATES

**COMPARING MEASUREMENTS IN THE PRINCIPAL FRAME AND IN N-THEOREM COORDINATES**

*x* = Log10[*H**2**SO**4**, molecules cm**-3*]
*y* = Log10[*Organic, ppb*]
*z* = Log10[*J, cm**-3*]

***Equation of plane ***
***(rotated to {x, y, z} coords)****** ***

## PCA-BASED  REDUCTION TO HOMOMOLECULAR NUCLEATIONVIA THE  “AVERAGE VIRTUAL MONOMER”

**PCA-BASED  REDUCTION TO HOMOMOLECULAR NUCLEATION****VIA THE  “AVERAGE VIRTUAL MONOMER”**

***Kulmala and Viisanen, J. Aerosol Sci. 22, S97, 1991***

***Equation of line:***

**0.2 ppb p-toluic acid**

**0.4 ppb p-toluic acid**

**use reference concentrations**
**(e.g. centroid of data set)**

**use actual concentrations**

***slope***

## NON-ISOTHERMAL NUCLEATION THEOREM :

**NON-ISOTHERMAL NUCLEATION THEOREM :**
- SELECTION OF THE TEMPERATURE COORDINATE** **

**Kinetic extension of “2nd nucleation theorem” **
**(McGraw and Wu, 2003)**

**Recommended form when only vapor **
**composition is known:**
**No bulk or equilibrium vapor pressure**
**information required!**

**Integration of recommended form gives **
**structure like the Clapreyron relation for vp;**
**(prefactor tends to be constant in both cases). **

- Suggested temperature coordinate: 1000/*T *

## ANALYSIS OF SIMULATED WATER NUCLEATION DATA

**(BASED ON AN EMPIRICAL FIT TO ACTUAL RATE MEASUREMENTS)**

**220K**

**240K**

**250K**

**260K**

***Data and fit from: Wölk and Strey, J. Chem, Phys. B105, 11683, 2001***

**curves from data set**

**PCA planar  fit**
**based on 200 simulated data points**
**entering random values for *****T***
**and [H****2****O] into empirical fit of Wölk and Strey**

***PCA fits captures T-dependence well but misses small curvature and slope changes***

## REGRESSION ANALYSIS OF RESIDUALS

**(INCLUDING QUADRATIC TERMS)**

- 3 = a1+a2(1)2+ a3 1 2 + a4(2)2

*z *- *z*(planar fit) =
- b1+b2 x2+ b3 xy + b4 y2

***quadratic in principal coordinates***
***(greater accuracy)***

***quadratic in original coordinates***
***(more amenable to physical ***
***interpretation)***

**220K**

**220K**

**240K**

**250K**

**260K**

**240K**

**250K**

**260K**

## EXTENSION TO ATMOSPHERIC NEW PARTICLE FORMATION

**EXTENSION TO ATMOSPHERIC NEW PARTICLE FORMATION**

**** *****McGraw and Saunders, Aerosol Sci. and Technol. 3, 367-380***
***This paper considers all of these processes in an aerosol reactor study - with the exception of coagulation loss of nano-particles to background aerosol.***

**Our immediate goal is to build on this model**, incorporating the new multivariate nucleation analysis based on the N-theorem described here, and examine the sensitivity of 3nm and 10nm particle formation rates to atmospheric conditions.**

## SUMMARY

- •  Determined initiation mechanism for nucleation in the ternary system
- p-toluic acid/ sulfuric acid/ water directly from rate measurements
- (e.g. a single p-toluic acid molecule is sufficient to initiate the process).
- •  Introduced application of PCA and exploited PCA/NT synergism
- for analysis with flexible data acquisition.
- •  Developed compact NT-based parameterizations for nucleation rate
- in terms of vapor composition and temperature  (source term for
- new particle formation).
- •  Modification of “second nucleation theorem” to work directly with
- species concentrations rather than supersaturation.
- •  _Long term goals_:  Support interpretation of laboratory and ultimately -
- when aerosol is included -  field measurements.  Improve capability for
- prediction of new particle formation in climate models.

## ACKNOWLEDGEMENTS

- •  Renyi Zhang for sharing data and collaborations on the
-     N-Theorem/PCA data analysis and interpretation for the ternary system.
- •  The DOE Atmospheric Science Program for support.