text:
GENERAL TERMS & CONDITIONS FOR INTERNET SALES
General terms and conditions of sales
DEFINITIONS AND INTERPRETATIONS
Without having previously been defined, the terms used here in beginning with a capital letter have the meanings set out below:
"Client" means a natural person, above the age of 18, acting to satisfy his/her personal requirements and with the full legal capacity to commit himself/herself to these general terms and conditions.
"Terms & Conditions of sale for the rate booked" means the specific conditions for each booking made by the Client.
"Booking confirmation" means the document summarising the details of the booking made by the Client sent to the Client by the Website or Hotel.
"Booking request" means any booking request for a hotel room submitted by the Client.
"Hotel" means the Maison Albar Hotels L'Imperator hotel, operated by the simplified joint stock company (SAS) MAISON ALBAR IMPERATOR with capital of € 11,774,475.00, with its registered office at 96, boulevard Haussmann - 75008 PARIS, entered in the Paris Trade & Companies Resgister with the number 829 129 501.
"Partners" means all service providers, which have concluded a contract for the provision of services or a partnership agreement with the Hotel.
"Service" means any service for booking a hotel room used by the Client on the Hotel's Website.
"Hotel's Website" means the Hotel's dedicated website, which is accessible at the following address: https://www.maison-albar-hotels-l-imperator.com/en/
ARTICLE 1 - Scope :
These General Terms & Conditions of Sale apply, without restriction or reservation, to all purchases of services for booking hotel rooms and related services ("the Services") offered by the Hotel ("the Service Provider" or the "Hotel")) to non-business customers and clients ("The Clients or the Client") on the Hotel's Website.
The key characteristics of Services are presented on the Hotel's Website.
The Client is invited to acquaint himself/herself with them before making a booking. The choice of and purchase of a Service is the Client's sole responsibility.
The Client declares:
That he/she has the full legal capacity to commit himself/herself to these General Terms & Conditions of Sale.
That he/she is booking hotel rooms and related services for his/her own personal use.
That he/she is able to save and print these General Terms & Conditions of Sale.
The Hotel's details are as follows:
Maison Albar Hotels L'Imperator
These terms & conditions apply, to the exclusion of all other terms & conditions and, in particular, those applicable to other channels for marketing Services.
The Client is made aware that the Hotel concludes partnership agreements with third party travel providers, to enable him/her to search for, select and book rooms in the Hotel using the services provided by these partners on their websites. All hotel room bookings made under these conditions involve the Client reading and fully and unreservedly accepting the service provider's specific conditions, the terms & conditions of sale for the rate booked and these general terms & conditions of sale. The Client declares that he/she has obtained from the Hotel all the necessary and available information on the Website.
These General Terms & Conditions of Sale are accessible at any time on the Hotel's Website and, where applicable, shall take precedence over any other contradictory version or document.
Unless proven otherwise, the data stored in the service provider's IT system constitutes proof of all transactions concluded with the Client. Thus, the entry of bank details, acceptance of the General Terms & Conditions of Sale, the Terms & Conditions of Sale for the rate or the Booking request, has the same legal effect, between the Hotel and the Client, as a handwritten signature on a physical document. The computerised records stored within the Hotel's IT systems shall be kept in reasonably secure conditions and regarded as proof of communication, orders and payments involving the Hotel and the Client.
The Hotel shall retain documents confirming the conclusion of a contract in an electronic or printed form for a maximum of 5 years.
The Client is informed that his/her IP address is logged when making a booking.
Pursuant to the RGPD, the Client has a right to access, correct and object to all of his/her personal data, at any time, by writing a letter, with proof of his/her identity, to:
These General Terms & Conditions of Sale also comprise the Personal Data Protection Charter.
The Client declares that he/she has read these General Terms & Conditions of Sale (including the Personal Data Protection Charter) and has accepted them by ticking the box intended for this purpose, before completing the online booking procedure, as well as the General Terms & Conditions for Using the Hotel's Website.
Confirmation of the booking of Services by the Client implies the unrestricted and unreserved acceptance of these General Terms & Conditions of Sale (including the Personal Data Protection Charter).
The Client confirms that he/she has the necessary capacity to enter into a contract and purchase the Services offered in the Hotel's Website.
As these General Terms & Conditions of Sale may be subject to subsequent amendment, the applicable version for the Client's purchase is that appearing on the website on the booking date.
These General Terms & Conditions of Sale are applicable for the period for which the services offered by the Hotel are online on the Hotel's Website. The Hotel reserves the right to temporarily or permanently close access to its Website.
ARTICLE 2 - Bookings :
The Client selects the services he/she wishes to book on the Website, as follows:
He/she selects the type of room and the rate
He/she selects related services, where applicable
He/she checks and confirms the booking details, the total cost of the booking and the terms & conditions for the rate
He/she enters his/her details
He/she enters his/her debit/credit card details as a guarantee or for prepayment
He/she reads and accepts the general terms & conditions of sale and the terms & conditions for the rate selected before confirming his/her booking
He/she confirms his/her booking
The Client confirms that he/she is aware of the nature, purpose and procedures for booking Services offered by the Hotel and has requested and obtained the necessary information to proceed with his/her booking in full knowledge of the facts. He/she is solely responsible for his/her choice of services and that they are commensurate with his/her requirements, such that the Hotel may not be held liable in this respect.
The Client undertakes to provide the information requested on the Booking request and confirms the truthfulness and accuracy of the information provided.
The contractual information is presented in French and forms the subject of a confirmation, at the latest when the booking is confirmed by the Client.
For bookings made solely on the internet, a booking is recorded on the Service Provider's website when the Client accepts these General Terms & Conditions of Sale by ticking the box intended for this purpose and confirming his/her booking. The Client has the opportunity to check the details of his/her booking and its total price, and to correct possible errors, before confirming his/her acceptance. This confirmation implies acceptance of the entire General Terms & Conditions of Sale and constitutes proof of a contract of sale.
Therefore, it is the responsibility of the Client to check the accuracy of the booking and to immediately report any errors.
The sale of Services shall only be considered definitive once the Client has been sent a confirmation that the booking has been accepted by the Service Provider, by e-mail, and after the full price has been received by the latter.
All bookings made on the Hotel's Website constitute a contract concluded remotely between the Client and the Service Provider.
The Hotelreserves the right to cancel or refuse any booking from a Client with whom there may be a dispute relating to payment for a previous booking.
Every booking is made in the name of a Client and may not transferred to a third party under any circumstances.
Cancellation of a Flexible rate booking
In the event of the cancellation of a Flexible rate booking by the Client after it has been accepted by the Hotel but less than 48 hours before his/her arrival, for whatever reason, a sum equivalent to the total cost of the 1st night shall be rightfully collected by the Service Provider and charged to the Client, by way of compensation for the loss sustained.
Cancellation / Amendment of a Non-cancellable Non-refundable rate booking
In the event of the cancellation or amendment of a Non-cancellable Non-refundable rate booking by the Client, for whatever reason, a sum equivalent to 100% of the total prepaid amount shall be rightfully collected by the Service Provider and charged to the Client. The same shall apply should the Client fail to appear on his/her intended arrival date.
ARTICLE 3 - RATES :
The Services offered by the Service Provider are provided at the rates appearing on the Hotel's Website when the booking is recorded by the Service Provider. The prices are in Euros, excluding VAT and including VAT.
The rates take account of possible reductions that may be granted by the Service Provider under the conditions specified on the Hotel's Website.
These rates are fixed and non-adjustable for their validity period, as stipulated on the Hotel's Website, with the Service Provider reserving the right to amend the prices at any time, outside this validity period.
The rates are displayed before and at the time of the booking made by the Client. They apply per room for the selected number of guests and date.
The rates are confirmed to the Client as a figure including VAT (excluding Occupancy taxes) in the currency used by the Hotel. They include VAT at the applicable rate on the day of the booking; any changes in the applicable VAT rate will be automatically applied to the stated rates on the billing date. The same shall apply to all amendments or introductions of new legal or statutory taxes imposed by the relevant authorities.
The rates do not include Occupancy Tax, which is payable locally to the Hotel.
The Client undertakes to pay these various taxes to the Hotel, without disputing them.
The payment requested from the Client corresponds to the total value of the purchase, with the exception of this Tax.
Unless stipulated otherwise on the Website, related services (breakfast, etc.) are not included in the price.
Conversions into foreign currency are provided for information and are non-contractual. Should a rate involve a direct payment to the Hotel on the Client's arrival or departure and should the Client's currency not be the same as that used by the Hotel, the rate charged by the Hotel may differ from that stipulated at the time of booking, as the result of changes in the exchange rate between the booking date and the payment date.
A bill is produced by the Service Provider and issued to the Client when the Services booked are provided.
ARTICLE 4 - PAYMENT TERMS :
In the case of immediate payment on the day of booking (Non-cancellable Non-refundable rate)
The price is payable immediately, in full on the day on which the booking is confirmed by the Client, in accordance with the procedures outlined in the article entitled "Bookings" above, using a secure payment method:
- using a debit/credit card: Visa, Mastercard, American Express, Cup, Diners Club card, JCB card, other credit cards (e-credit card).
When making the booking, the Client provides his/her bank details by stating the name on the debit/credit card, the card number and the expiry date (the debit/credit card must be valid until the final day of the stay) and the security code.
Payment data is exchanged in encrypted form using the SSL protocol.
On check-in, the Client shall present the Hotel with the debit/credit card that he/she used to pay for the booking. He/she may be asked to provide a form of identification as part of debit/credit card fraud prevention procedures.
The Service Provider shall not be obliged to provide Services ordered by the Client if the price has not been paid in full under the conditions specified above.
Payments made by the Client shall not be considered definitive until the amounts due have actually been received by the Service Provider.
In the case of immediate payment when services are provided (Flexible Rate)
The price is payable immediately, in full on the day on which the Services booked are provided under the conditions specified in the article entitled "Provision of Services" below and as stipulated on the bill issued to the Client, using a secure payment method:
On arrival, the Hotel shall ask the Client to pay a deposit or to authorise a debit against his/her debit/credit card, for the purpose of guaranteeing the payment of amounts for services used at the Hotel.
The Client shall provide his/her bank details by stating the name on the debit/credit card, the card number and the expiry date (the debit/credit card must be valid until the final day of the stay) and the security code.
The Client may be asked to provide a form of identification as part of debit/credit card fraud prevention procedures.
ARTICLE 5 - Provision of Services :
Services booked by the Client, including hotel room booking services and related services, shall be provided as follows, under the conditions specified in these General Terms & Conditions of Sale, supplemented by the Terms & Conditions of Sale for the Rate, which the Client has read and accepted when making his/her booking on the Hotel's Website.
On arrival, the Client will be asked to provide a form of identification in order to fulfil his/her obligation to complete a Police Registration Form.
The Hotel is completely non-smoking. The client shall be liable for direct and/or indirect, consequential damage caused by smoking within the Hotel. As a result, the Client will be liable for the entire costs of cleaning and returning the damaged item or area to its original state.
Pets, which are kept on a leash or in a cage in the establishment's communal areas, may be accepted, depending on the Hotel's policy, on payment of a surcharge. For hygiene reasons, pets are not allowed in the Hotel's cafés and restaurants.
The Client is entirely responsible for his/her personal effects left in the Hotel room, in particular not in the safe, or in the Hotel's public areas. The Hotel may not be held responsible for the loss or theft of, or deterioration or damage to the aforementioned personal effects.
The client agrees and undertakes to use his/her room in a responsible manner. Any behaviour contrary to public decency and public order shall result in the Hotel asking the Client to leave the establishment without any compensation or reimbursement if a payment has already been made. In the event that no payment has yet been made, the Client must pay for the nights he/has stayed before leaving the establishment.
The Client shall be liable for all direct and/or indirect, consequential damage he/she has caused in the room booked or that he/she may cause within the Hotel. As a result, he/she undertakes to pay the Hotel the cost of the aforementioned damages, without prejudice to compensation that may be due, plus court costs and legal fees incurred by the Hotel.
WiFi access (payable or free) allowing clients to connect to the internet may be available, depending on the Hotel's policy. The Client undertakes to ensure that the IT resources placed at his/her disposal by the Hotel shall not be used in any way to reproduce, represent, provide or convey to the public, works or objects protected by a copyright or similar right, including texts, images, photographs, musical works, audiovisual works, software and video games, without the permission of the holders of rights set out in Volumes I and II of French Intellectual Property Code where this permission is required. Should the Client fail to comply with the above obligations, he/she risks being accused of a counterfeiting offence. The Client is also obliged to comply with the Hotel's internet access provider's security policy, including the rules for using the security resources deployed with the aim of preventing the illicit use of IT resources and to refrain from any acts impairing the effectiveness of these resources.
Unless explicitly stipulated otherwise, the Room will be made available to the Client on the day of his/her arrival at 3 p.m. and the Client shall vacate the room on the day of his/her departure at midday. Should he/she fail to do so, the Client shall be charged for an additional night. The Client must check his/her departure date. In the event of early departure, costs equivalent to one night shall be charged, except if the Client has advised the Hotel of his/her early departure at least 24 hours in advance.
The Service Provider undertakes to make every effort to provide the Services booked by the Client, within the framework of a best endeavours obligation.
The Client shall have a period of 8 days following the date of his/her departure from the Hotel to express reservations or complaints relating to the provision of Services, in writing, with all the supporting documents, to the Hotel.
No complaint can be validly accepted without the Client adhering to these formalities and deadlines.
Should the Client not have explicitly expressed reservations or complaints by this deadline for the receipt of Services, these shall be deemed compliant with the booking, in terms of quantity and quality.
In the event of relocation
In the case of an unforeseen event, a case of force majeure or it not being possible for the Hotel to provide the Client with the room booked, the Hotel reserves the right to, fully or partially, accommodate the Client in an equivalent category hotel, for services of the same kind and subject to the Client's prior agreement.
ARTICLE 6 – RIGHT OF WITHDRAWAL :
The Client does not benefit from the right of withdrawal, given the nature of the services provided.
Therefore, the contract is concluded definitively when the booking is made by the Client in accordance with the procedures specified in these General Terms & Conditions of Sale.
ARTICLE 7 - The Service Provider's liability - Guarantee :
The Service Provider shall protect the Client, in accordance with the statutory provisions and without additional payment, against any lack of compliance or hidden defects, originating from a failure to provide the Services booked and actually paid for under the conditions and in accordance with the procedures defined in these General Terms & Conditions of Sale.
The Services provided via the Hotel's Website comply with the applicable regulations in France. The Service Provider may not be held liable in the event of a failure to comply with legislation in the country in which the Services are provided, which it is the responsibility of the Client, who is solely responsible for the choice of the Services requested, to check.
ARTICLE 8 - Data protection :
In accordance with the GDPR, it should be noted that the personal data requested from the Client is necessary for processing his/her booking and for producing bills, in particular.
This data is processed and intended for the Hotel and may be passed on to its possible partners entrusted with implementing, processing, managing and dealing with payments for bookings, as well as the Client's stay.
In addition, the Hotel may send its clients its e-mail newsletter, promotional offers and/or a guest satisfaction questionnaire following their stay.
In accordance with the applicable national and European regulations, the Client has a permanent right to access, amend, correct and object to information held about him/her.
This right can be exercised under the conditions and in accordance with the procedures specified on the Hotel'sWebsite.
The Hotel's data protection policy can be viewed within the Personal Data Protection Charter, which is available on the Hotel's Website.
ARTICLE 9 - Intellectual property :
The content of the Hotel's Website is the property of the Vendor and its partners and is protected by French and international laws relating to intellectual property.
Any total or partial reproduction of this content is strictly prohibited and may constitute a counterfeiting offence.
ARTICLE 10 - Unforeseeability :
These General Terms & Conditions of Sale expressly exclude the legal system of unforeseeability for all service operations provided by the Service Provider to the Client. Therefore, the Service Provider and the Client each waive the right to invoke the system of unforeseeability stipulated therein, undertaking to fulfil their obligations even if the contractual balance is disrupted by circumstances that were unforeseeable when concluding the sale, even if their performance should prove to be excessively burdensome, and to bear the economic and financial consequences of this.
ARTICLE 11 - Force majeure :
The Parties may not be held responsible if the non-fulfilment or late fulfilment of any one of their obligations, as described in these General Terms & Conditions of Sale, results from a case of force majeure.
Article 12: MISCELLANEOUS PROVISIONS :
The General Terms & Conditions of Sale, the Personal Data Protection Charter, the Conditions of sale for the rate booked by the Client, the Booking request and Confirmation of the booking by the Client, constitute the entire agreement between the parties within the limits of its purpose. As a result, to this extent, they supersede all previous verbal or written agreements.
No tolerance, of whatever kind, extent, duration or frequency, may be considered as creating any right whatever and may not limit in any way, the opportunity to invoke each of the clauses of these General Terms & Conditions of Sale, at any time, without any restrictions.
Any clause of these General Terms & Conditions of Sale, which may be declared invalid or unlawful by a competent judge, shall be rendered unenforceable, but its invalidity shall not undermine the other clauses, or affect the validity of the General Terms & Conditions of Sale as a whole or their legal effects.
ARTICLE 13 - Applicable law - Language :
These General Terms & Conditions of Sale and the resulting transactions are governed by and subject to French law.
These General Terms & Conditions of Sale are written in English. Should they be translated into one or more foreign languages, the English text will be the only authoritative version in the event of a dispute.
ARTICLE 14 - DISPUTES :
All disputes to which the purchasing and sale transactions concluded in accordance with these General Terms & Conditions of Sale may give rise, relating to their validity, interpretation, performance, termination, their consequences and effects, and which it may not be possible to resolve between the Hotel and the Client shall be heard by the competent courts under the conditions of common law.
The Client is informed that, in all cases, he/she may make use of conventional mediation services, in particular provided by the Consumer Mediation Commission if exists or any alternative method for resolving differences (conciliation, for example) in the event of a disputes.
ARTICLE 15 – PRE-CONTRACTUAL INFORMATION - Acceptance by the Client :
The Client confirms that, prior to making his/her booking and on concluding the contract, it has received these General Terms & Conditions of Sale, in a legible and comprehensible form, and specifically the following information:
the essential characteristics of Services, taking account of the communication medium used and the Service in question,
the price of Services and ancillary costs,
in the absence of immediate performance of the contract, the date or deadline by which the Service Provider undertakes to provide the Services booked,
information relating to the identity of the Service Provider, its postal address, telephone number and e-mail address, and its activities, if these are not apparent from the context,
information relating to legal and contractual guarantees and the procedures for implementing them,
digital content functionalities and, where applicable, its interoperability,
the option of making use of conventional mediation services in the event of a dispute,
information relating to important contractual conditions.
the payment methods accepted.
The act, by a natural person (or legal entity), of making a booking via the Hotel's Website implies the full and complete adherence to and acceptance of these General Terms & Conditions of Sale and the obligation to pay for the Services ordered, which is expressly acknowledged by the Client who, specifically, waives the right to invoke any conflicting document, which would be unenforceable as regards the Service Provider.


"Believe what you read. Teach what you believe. Practice what you teach".
This homily is the first part of a two part homily I preached this summer. The second part follows.
Do you remember last's week's gospel reading? What was it about? (I bet you did not expect a pop quiz at church today!) It was….The miracle of the multiplication of the bread, which happens to be the first few lines of the 6 chapter in the Gospel according to John.
Now Father might have mentioned this last week but starting last Sunday and for a period of 5 Sundays the Gospel reading will be snippets from this chapter, so you can bet that if the Church spends 5 weeks in just one chapter of scripture it must be a pretty important one right? It is so important that although broken in 5 parts all these readings must be taken as one long story that I hope you will take time to read during the week.
It is clear that Jesus can read what is in their hearts. He knows they do not care about his miracles; the only reason why they went to the trouble of crossing the lake was because they wanted more food! Which is understandable because at the times of Jesus food was not easy to find. And here Jesus gives them the first lesson of the day: he tells them "do not trouble yourself searching for normal food" Seek spiritual food instead, my words, the signs and wanders I do… these things are good for your soul…these will sustain you for all eternity. This is food that only the one sent by God the Father can provide.
They do not believe him! Jesus is telling them how to get this food and they go back to wanting him to proof who he is by… Feeding them again! The only thing in their minds was the free meal!
Imagine the surprise of the people listening to Jesus when he uttered these words. For us Catholics to hear Jesus call himself the bread of life is second nature, but to them, for these people that were expecting perhaps some kind of special bread, to hear Jesus call himself The Bread of Life must have been very confusing! I'm sure many of then though "he could not possibly be talking about heating his flesh and drinking his blood". But we all know what he meant. He IS the bread of life that comes down from heaven and gives eternal life to the world! And the only thing we have to do is believe in his words! And up to this date we express this belief by eating His real flesh and drinking his real blood in the Eucharist at every mass in every Catholic Church in the world.


Wouldn't It Be Great To Have Enough Doctors for All Ukiahans?
In Ukiah, we do not have enough primary care doctors to care for everyone. Primary care is the term used to describe whole person health care, not specialized care focused on a certain body system or disease. Within primary care, there are different specialties, including family medicine, internal medicine, and pediatrics. Primary care doctors are the doctors you go to first when you're sick, if you're lucky enough to have one.
Family medicine is unique in that it is the only specialty that does not narrow its practice by organ/system, age, or gender, according to one of the nationally (and internationally) known leaders in family medicine who is interested in little 'ole Ukiah, professor Richard Roberts, MD, JD. A pediatrician, for example, focuses on children. An internist focuses exclusively on adults. An obstetrician/gynecologist may provide some primary care, but only treats women.
In Ukiah, we have far fewer primary care doctors than we need, and many of them are getting close to retirement. Chief of Staff Dr. Charlie Evans, a family physician who works in the Ukiah Valley Medical Center (UVMC) Emergency Department, shared startling news when he said, "We need 42 primary care doctors, and we have 28—9 of whom are over 65." This equates to approximately 40,000 local people who don't have a primary care doctor, and who therefore put off care until they either need emergency care or until their disease progresses to the point where treatment is no longer effective or available. Clearly, we need to do something.
This is where the residency program comes in. After doctors finish medical school, they are trained in a specialty, like family medicine. Statistics show that a huge percentage of residents (50-80 percent) stay close to wherever they finish their residency. These new doctors get comfortable with the area, build relationships, and their children get established in local schools.
Since then, the hospital and its parent company, Adventist Health, have agreed to house the program, and offered a $100,000 matching grant to go toward the $1.5 million start-up costs. A group of community-minded locals called Family Medicine Education for Mendocino County (FMEMC) have been busy publicizing and fundraising. Once the residency program is up and going, it will be self sustaining because it will qualify for federal funding after its first year—four years from now.


text:
\section{\label{sec:sce1}Introduction}
Nucleation is a very basic phenomena which plays a vital role in various material processing in industry ranging from steel production to food and beverage industries~\cite{Kelton2010}.  Recently, researchers have focused on the nucleation of complex materials which are relevant to our daily life~\cite{Kelton2010}.  The nucleation of such complex materials can also be complex and may involve intermediate metastable phases~\cite{Ostwald1897,Vekilov2010, Gebauer2008,Chung2009}.  These metastable intermediate phases indicate a composite nucleus that consists of a core of stable new phase surrounded by a wetting layer~\cite{Vekilov2010} of intermediate phase, which is nucleated from the metastable parent phase.

The author~\cite{Iwamatsu2011} recently examined the thermodynamics and free-energy surface of a critical nucleus using capillarity approximation when an intermediate metastable phase exists  The author discovered the possibility of such a composite critical nucleus at the saddle point of the free-energy surface~\cite{Iwamatsu2011}. In order to study the free-energy surface of nucleation, we utilized a three-phase system and assumed the existence of an intermediate metastable phase in addition to an unstable parent phase and a stable new phase.  We chose a composite binary nucleus that consists of the stable new phase and the metastable intermediate phase.  This binary nucleus is completely phase separated, and the metastable phase is segregated on to the surface.  By studying the free-energy surface of the two-component system in the two-dimensional space~\cite{Iwamatsu2011}, we can easily visualize the free-energy surface and are able to locate the saddle point that corresponds to the critical nucleus.  Even though such a composite nucleus has long been predicted theoretically~\cite{Kashchiev1998,Kashchiev2005,Meel2008}, it has been proved explicitly that the composite nucleus corresponds to the saddle point only through the use of a computer simulation~\cite{tenWolde1997}.   Our calculations~\cite{Iwamatsu2011} clearly demonstrate that by using an analytical model the composite nucleus really corresponds to the saddle point of the free-energy surface.  A similar composite nucleus at the saddle point of the free-energy surface is considered in the problem of deliquescence~\cite{Djikaev2002,Shchekin2008,McGraw2009}.

From the free-energy surface, the author could easily calculate the so-called minimum-free-energy path (MFEP)~\cite{Iwamatsu2009}.  However, it has been recognized for more than thirty years that the steepest-descent direction on the free-energy surface~\cite{Reiss1950} that corresponds to the MFEP does not necessarily indicate the real nucleation pathway in binary nucleation~\cite{Stauffer1976,Temkin1984,Greer1990}.  This is because the asymmetry of the reaction rate will deflect the direction of nucleation flux from the steepest-descent direction. Likewise, the free-energy surface alone may not determine the nucleation pathway for the composite nucleus.  The reason for this is that the asymmetry between the reaction rate from the metastable parent phase to the intermediate metastable phase and that from the intermediate to stable phase will play some role to determine not only the nucleation rate but the direction of nucleation flux.  Therefore, the MFEP determined by the steepest-descent direction of free-energy surface~\cite{Iwamatsu2009} will not necessarily correspond to the {\it real} nucleation pathway.

This study focuses on the steady-state nucleation rate and flux at the saddle point for the composite nucleus through the theory of binary nucleation~\cite{Trinkaus1983,Wilemski1999}.  We must first determine the Fokker-Planck or the Zeldovich-Frenkel equation~\cite{Kelton2010,Wu1997,Zeldovich1943,Frenkel1955} from the Master equation for the composite nucleus (Section \ref{sec:sec2}).  The  theory of nucleation rate and flux for the binary nucleation when the reaction rate matrix is non-diagonal is re-constructed in Section~\ref{sec:sec3} and in the Appendix A and B as the theory of Trinkaus~\cite{Trinkaus1983} is too condensed to study various aspects of nucleation of multicomponent systems.  In section \ref{sec:sec4} the theory developed in Sections \ref{sec:sec2} and \ref{sec:sec3} will be applied to the composite nucleus.  An example to illustrate the process will be provided in order to show how the reaction rate will influence the nucleation rate and the direction of nucleation flux at the saddle point for the composite nucleus.  Section \ref{sec:sec5} will contain the conclusion of the study.

\section{\label{sec:sec2}Fokker-Planck equation for the composite nucleus}

In order to study the nucleation kinetics of the composite nucleus, the model shown in Fig.~\ref{fig:1x} is considered.  The model consists of a core of the stable new phase (number of molecules $n_1$) surrounded by an intermediate metastable phase (number of molecules $n_2$) nucleated in the metastable parent phase~\cite{tenWolde1997, Meel2008, Vekilov2010, Iwamatsu2011}.  Nucleation rates $\kappa^{+}$ and $\kappa^{-}$ are the reaction rates between the stable new phase and the metastable intermediate phase.  Nucleation rates $\alpha^{+}$ and $\alpha^{-}$ are the reaction rate or attachment rate from the metastable parent phase to the intermediate metastable phase.

A similar model has been used to study the partitioning transformation~\cite{Russell1968,Kelton2000,Diao2008} where the coupling of the interfacial and long-range diffusion fluxes is considered to study the time-dependent precipitate nucleation~\cite{Kelton2003}, void formation in irradiated metals~\cite{Russell1978}, and nanocrystal formation from metallic glasses~\cite{Kelton1998}.  In these linked-flux models~\cite{Russell1968,Kelton2000,Diao2008}, in contrast to our model~\cite{Iwamatsu2011}, the metastable intermediate phase does not really represent the thermodynamics phase.  Rather, it models a transition zone due to the diffusion of monomers~\cite{Russell1968, Kelton2000}.  Therefore, the nucleation rate $\alpha^{+}$ does not represents the attachment rate of monomer but is simply a parameter which approximately represents diffusion.  In our model, however, the intermediate phase represents the metastable thermodynamics phase and $\alpha^{+}$ is the attachment rate of the monomer.

\begin{figure}[htbp]
\begin{center}
\includegraphics[width=0.65\linewidth]{fig1.EPS}
\end{center}
\caption{
A composite critical nucleus model that consists of a stable new phase (number of molecules $n_1$) surrounded by an intermediate metastable phase (number of molecules $n_2$) nucleated in the metastable parent phase~\cite{Iwamatsu2011}.  Nucleation rates $\kappa^{+}$ and $\kappa^{-}$ are the reaction rates between the stable new phase and the metastable intermediate phase.  Nucleation rates $\alpha^{+}$ and $\alpha^{-}$ are the reaction rate or attachment rate from the metastable parent phase to the intermediate metastable phase.   }
\label{fig:1x}
\end{figure}

The Master equation for the time-dependence of the number of clusters $f\left(n_1,n_2,t\right)$ that consists of $n_1$ molecules of the stable phase in the core and $n_2$ molecules of the intermediate metastable phase  in the surrounding wetting layer is written generally~\cite{Kelton2010,Wu1997, Reiss1950} in the form
\begin{equation}
\frac{\partial f\left(n_1,n_2,t\right)}{\partial t}=\left(-J_{n_1}+J_{n_1-1}\right)
+\left(-J_{n_2}+J_{n_2-1}\right),
\label{eq:2z}
\end{equation}
where
\begin{eqnarray}
J_{n_1}&=&\kappa^{+}\left(n_1,n_2\right)f\left(n_1,n_2,t\right) \nonumber \\
&-&\kappa^{-}\left(n_1+1,n_2-1\right)f\left(n_1+1,n_2-1,t\right), \nonumber \\
J_{n_1-1}&=&\kappa^{+}\left(n_1-1,n_2+1\right)f\left(n_1-1,n_2+1,t\right) \nonumber \\
&-&\kappa^{-}\left(n_1,n_2\right)f\left(n_1,n_2,t\right),
\label{eq:3z} \\
J_{n_2}&=&\alpha^{+}\left(n_1,n_2\right)f\left(n_1,n_2,t\right) \nonumber \\
&-&\alpha^{-}\left(n_1,n_2+1\right)f\left(n_1,n_2+1,t\right), \nonumber \\
J_{n_2-1}&=&\alpha^{+}\left(n_1,n_2-1\right)f\left(n_1,n_2-1,t\right) \nonumber \\
&-&\alpha^{-}\left(n_1,n_2\right)f\left(n_1,n_2,t\right). \nonumber
\end{eqnarray}
Using the detailed balance condition,
\begin{eqnarray}
\kappa^{-}\left(n_1+1,n_2-1\right)&=&\kappa^{+}\left(n_1,n_2\right)\frac{f_{\rm eq}\left(n_1,n_2\right)}{f_{\rm eq}\left(n_1+1,n_2-1\right)}, \nonumber \\
\alpha^{-}\left(n_1,n_2+1\right)&=&\alpha^{+}\left(n_1,n_2\right)\frac{f_{\rm eq}\left(n_1,n_2\right)}{f_{\rm eq}\left(n_1,n_2+1\right)},
\label{eq:4z}
\end{eqnarray}
where the equilibrium cluster distribution $f_{\rm eq}\left({\bm n}\right)$ is given by the usual Boltzmann distribution
\begin{equation}
f_{\rm eq}\left({\bm n}\right)=f_{0}\exp\left(-\beta G\left({\bm n}\right)\right),
\label{eq:5z}
\end{equation}
and $G\left({\bm n}\right)$ is the work of cluster formation for a cluster with composition ${\bm n}=\left(n_1,n_2\right)$ and $\beta$ is the inverse temperature.   Then, Eq.~(\ref{eq:2z}) can be written as
\begin{eqnarray}
&&\frac{\partial f\left(n_1,n_2,t\right)}{\partial t}= \nonumber \\
&-&\kappa^{+}\left(n_1,n_2\right)f_{\rm eq}\left(n_1,n_2\right) \nonumber \\
&&\times\left[\frac{f\left(n_1,n_2,t\right)}{f_{\rm eq}\left(n_1,n_2\right)}-\frac{f\left(n_1+1,n_2-1,t\right)}{f_{\rm eq}\left(n_1+1,n_2-1\right)}\right]
\nonumber \\
&+&\kappa^{+}\left(n_1-1,n_2+1\right)f_{\rm eq}\left(n_1-1,n_2+1\right) \nonumber \\
&&\times\left[\frac{f\left(n_1-1,n_2+1,t\right)}{f_{\rm eq}\left(n_1-1,n_2+1\right)}
-\frac{f\left(n_1,n_2,t\right)}{f_{\rm eq}\left(n_1,n_2\right)}\right] \nonumber \\
&-&\alpha^{+}\left(n_1,n_2\right)f_{\rm eq}\left(n_1,n_2\right) \nonumber \\
&&\times\left[\frac{f\left(n_1,n_2,t\right)}{f_{\rm eq}\left(n_1,n_2\right)}
-\frac{f\left(n_1,n_2+1,t\right)}{f_{\rm eq}\left(n_1,n_2+1\right)}\right] \nonumber \\
&+&\alpha^{+}\left(n_1,n_2-1\right)f_{\rm eq}\left(n_1,n_2-1\right) \nonumber \\
&&\times\left[\frac{f\left(n_1,n_2-1,t\right)}{f_{\rm eq}\left(n_1,n_2-1\right)}
-\frac{f\left(n_1,n_2,t\right)}{f_{\rm eq}\left(n_1,n_2\right)}\right].
\label{eq:6z}
\end{eqnarray}
In the continuum limit, we have
\begin{equation}
\frac{\partial f\left({\bm n}\right)}{\partial t}=-\left(\frac{\partial J_{n_1}}{\partial n_1}+\frac{\partial J_{n_2}}{\partial n_2}\right)=-{\rm div}{\bm J},
\label{eq:10z}
\end{equation}
where the components of the nucleation flux $\bm J$ are given by
\begin{eqnarray}
J_{n_1}&=-&\kappa^{+}f_{\rm eq}\left\{\frac{\partial \Phi}{\partial n_1}
-\frac{\partial \Phi}{\partial n_2}\right\},  \label{eq:11z}
\\
J_{n_2}&=&-\alpha^{+} f_{\rm eq}\left\{\frac{\partial \Phi}{\partial n_2}\right\}  \nonumber \\
&&-\kappa^{+}f_{\rm eq}\left\{-\frac{\partial \Phi}{\partial n_1}
+\frac{\partial \Phi}{\partial n_2}\right\},
\label{eq:12z}
\end{eqnarray}
and
\begin{equation}
\Phi\left(n_1,n_2,t\right)=\frac{f\left(n_1,n_2,t\right)}{f_{\rm eq}\left(n_1,n_2\right)},
\label{eq:9z}
\end{equation}
and simplified notations $\kappa^{+}=\kappa^{+}\left(n_1,n_2\right)$, $\alpha^{+}=\alpha^{+}\left(n_1,n_2\right)$ and $f_{\rm eq}=f_{\rm eq}\left(n_1,n_2\right)$ and $\Phi=\Phi\left(n_1,n_2,t\right)$ are used.  Equation (\ref{eq:10z}) is called the Fokker-Planck~\cite{Risken1989} or the Zeldovich-Frenkel equation~\cite{Zeldovich1943,Frenkel1955}.

Equations (\ref{eq:11z}) and (\ref{eq:12z}) can be put in the form of a matrix equation
\begin{equation}
{\bm J}_{\bm n}=-f_{\rm eq}\left({\bm n}\right){\bm R\left({\bm n}\right)}{\bm \nabla}_{\bm n}\Phi\left({\bm n}\right),
\label{eq:15z}
\end{equation}
where $\bm J$ and ${\bm \nabla}_{\bm n}$ are row vectors, and $\bm R$ is a symmetric square matrix defined through Eqs.~(\ref{eq:11z}) and (\ref{eq:12z}).  The two fluxes $J_{n_1}$ and $J_{n_2}$ are linked by the non-diagonal rate matrix $R$. Eq.~(\ref{eq:15z}) is originally derived by Russell~\cite{Russell1968} though the analysis of the nucleation flux at the saddle point was mostly qualitative and numerical and did not present various analytical formulas which will be derived in this paper.

A general theory of barrier crossing by the decay of metastable state has been formulated by Langer~\cite{Langer1969} using the Fokker-Planck equation more than forty years ago by generalizing the Kramers' theory~\cite{Kramers1940} of barrier crossing, and has been extended by several workers~\cite{Berezhkovskii2005, Peters2009} to study the reaction and nucleation pathways in various problems not limited to binary nucleation.  This Kramers-Langer-Berezhkovskii-Szabo (KLBS) theory~\cite{Kramers1940,Langer1969,Berezhkovskii2005} considers a general problem of escape probability from a potential well of the metastable state.  In this KLBS theory, the nucleation flux ${\bm J}$ is the probability flux and $\Phi$ is called the Kramers crossover function~\cite{Peters2009} or the commitment or splitting probability~\cite{Rhee2005} which represents the probability to cross the barrier from the given point in the phase space.

In this paper, we adopt the theory of binary nucleation~\cite{Reiss1950,Stauffer1976,Trinkaus1983,Temkin1984,Greer1990,Wu1997,Wilemski1999} since it does not need to characterize the initial metastable state by a potential well.  Also, the connections to the original Master equation are more transparent since the elementary process of nucleation is a discrete process of attachment and detachment of monomers.  It is well recognized~\cite{Stauffer1976} that we must return to the original discrete Master equation from the Fokker-Planck equation when we study the nucleation pathway of post-critical nucleus.  In any case, the formula based on KLBS theory~\cite{Kramers1940,Langer1969,Berezhkovskii2005} reduces to that used in the theory of binary nucleation~\cite{Trinkaus1983} when the thermal distribution in the metastable potential well is incorporated~\cite{Zitserman1990}.  We will extend the theory~\cite{Reiss1950,Stauffer1976,Trinkaus1983,Temkin1984,Greer1990,Wu1997,Wilemski1999} developed for binary nucleation to study the nucleation of composite nucleus in the next section.

\section{\label{sec:sec3} Magnitude and direction of steady state nucleation flux at the saddle point}

From this point, we will concentrate on the steady state nucleation flux at the saddle point ${\bm n}^{*}$ so that the nucleation flux is a time independent constant vector (${\rm div}{\bm J}=0$) and the rate matrix ${\bm R}={\bm R}\left({\bm n}^{*}\right)$ is also a constant matrix at the saddle point.

The general theory of nucleation flux of binary nucleation will be reformulated when the rate matrix $\bm R$ is non-diagonal so that we can study the nucleation of composite nucleus.   Most of the previous authors who studied the nucleation flux of binary nucleation have studied the cases where the rate matrix is diagonal~\cite{Stauffer1976,Wilemski1999}.  To exemplify this, we introduce the rotation matrix of the coordinate given by
\begin{equation}
{\bm V}\left(\theta\right)=
\begin{pmatrix}
\cos\theta & -\sin\theta \\
\sin\theta  &  \cos\theta
\end{pmatrix},
\label{eq:17z}
\end{equation}
which is the orthogonal transformation in vector space, and study the steady-state solution $\partial f/\partial t=0$ of the Fokker-Planck equation (\ref{eq:10z}) given by
\begin{equation}
{\bm \nabla}_{\rm n}^{\rm T}{\bm J}=\partial_{n_1}J_{n_1}+\partial_{n_2}J_{n_2}=0,
\label{eq:19z}
\end{equation}
where we have used a column vector ${\bm \nabla}_{\rm n}^{\rm T}$ for the differential operator ${\rm div}$ and the superscript T which means the transpose.

Using the rotation matrix in Eq.~(\ref{eq:17z}),  we can introduce a new coordinate ${\bm e}_{\bm \eta}$ whose origin is displaced at the saddle point ${\bm n}^{*}$ and rotate the unit vectors ${\bm e}_{n_1}$ and ${\bm e}_{n_2}$ by an angle $\theta$ through
\begin{equation}
{\bm e}_{\bm n}={\bm V}\left(\theta\right){\bm e}_{\bm \eta},\;\;\;
\mbox{or}\;\;\;\;
{\bm e}_{\bm \eta}={\bm V}^{\rm T}\left(\theta\right){\bm e}_{\bm n}.
\label{eq:21z}
\end{equation}
Then the displacement (column) vector from the saddle point ${\bm n}^{*}$
\begin{equation}
\Delta {\bm n}={\bm n}-{\bm n}^{*}=
\begin{pmatrix}
\Delta n_1 \\
\Delta n_2
\end{pmatrix},
\label{eq:22z}
\end{equation}
will be transformed into the displacement $\bm \eta$ through
\begin{equation}
\Delta{\bm n}={\bm V}\left(\theta\right){\bm \eta},\;\;\;
\mbox{or}\;\;\;\;
{\bm \eta}={\bm V}^{\rm T}\left(\theta\right)\Delta{\bm n}.
\label{eq:24z}
\end{equation}
The direction of nucleation flux given by Eq.~(\ref{eq:15z}) can be studied by inspecting the appropriate rotation of coordinate.

\subsection{\label{sec:3-1}Direction of steepest-descent of free-energy surface}

The steepest-descent direction of the free-energy surface, which is the direction of minimum free-energy path (MFEP)~\cite{Iwamatsu2009}, at the saddle point can be studied by expanding the work of cluster formation $G\left({\bm n}^{*}\right)$ around the saddle point ${\bm n}^{*}$ as~\cite{Reiss1950}
\begin{equation}
G\left({\bm n}\right)=G^{*}+\frac{1}{2}\Delta{\bm n}^{\rm T}{\bm G}\Delta{\bm n}
\label{eq:25z}
\end{equation}
using the displacement (column) vector defined in Eq.~(\ref{eq:22z}), where
\begin{equation}
{\bm G}=
\begin{pmatrix}
G_{11} & G_{12} \\
G_{21} & G_{22}
\end{pmatrix},
\label{eq:26z}
\end{equation}
with
\begin{equation}
G_{ij}=\frac{\partial^{2} G}{\partial n_i \partial n_j},
\label{eq:27z}
\end{equation}
is the Hessian matrix of the free energy at the saddle point ${\bm n}^{*}$, and $G^{*}=G\left({\bm n}^{*}\right)$.  By using the rotation matrix ${\bm V}\left(\theta\right)$ defined in Eq.~(\ref{eq:17z}),
we can diagonalize the matrix $\bm G$ and obtain the negative eigenvalue that corresponds to the steepest-descent direction as
\begin{equation}
\lambda_1 = \left(\left(G_{11}+G_{22}\right)-\sqrt{\left(G_{11}+G_{22}\right)^2-4{\rm det}{\bm G}}\right)/2,
\label{eq:30z}
\end{equation}
where ${\rm det}{\bm G}=G_{11}G_{22}-G_{12}^{2}<0$ at the saddle point.

The rotation angle $\theta$ is chosen such that $\lambda_{1}$ becomes negative and $\lambda_2$ becomes positive.  Then the rotation angle $\theta$ is given by
\begin{equation}
\tan \theta=\frac{\left(G_{22}-G_{11}\right)}{2G_{12}}
\pm\sqrt{\left(\frac{G_{22}-G_{11}}{2G_{12}}\right)^2+1},
\label{eq:32z}
\end{equation}
where $+$ sign must be chosen when $G_{12}<0$ otherwise $-$ sign must be chosen.  The steepest-descent direction is the direction of $\eta_1$ axis defined through
from Eq.~({\ref{eq:21z}) with $\theta$ given by Eq.~(\ref{eq:32z}).  This is the direction of the eigenvector of negative eigenvalue $\lambda_1$ of the matrix $\bm G$.

\subsection{\label{sec:3-2}Direction of gradient of $\Phi$}

Next, we will consider the direction of the gradient of $\Phi$, which has been studied intensively by Wilemski~\cite{Wilemski1999} and others~\cite{Wyslouzil1999,Li2000}.  This function $\Phi$ is also known as the Kramers crossover function~\cite{Peters2009} or the commitment or splitting probability~\cite{Rhee2005} in the general theory of barrier crossing, where $\Phi$ plays the role of the probability to cross the barrier.  We will extend the work of Wilemski~\cite{Wilemski1999} since we are dealing with nucleation by an attachment and detachment process.  Now, we rotate $\left(\Delta n_1,\Delta n_2\right)$ axis by $\omega$ and introduce a new coordinate $\left(\zeta_1,\zeta_2\right)$ through
\begin{equation}
{\bm \zeta} = {\bm V}^{\rm T}\left(\omega\right)\Delta_{\bm n}.
\label{eq:35z}
\end{equation}
Then, the component $J_{\zeta_1}$ and $J_{\zeta_2}$ are related to $J_{n_1}$ and $J_{n_2}$ through
\begin{equation}
{\bm J}_{\bm \zeta}={\bm V}^{\rm T}\left(\omega\right){\bm J}_{\bm n}.
\label{eq:37z}
\end{equation}
Similarly we obtain
\begin{equation}
{\bm \nabla}_{\bm \zeta}={\bm V}^{\rm T}\left(\omega\right){\bm \nabla}_{\bm n},
\label{eq:38z}
\end{equation}
which is the simplest transformation of covariant vector~\cite{Risken1989,Graham1977,Grabert1980} when the coordinate transformation is given by a linear orthogonal transformation Eq.~(\ref{eq:17z}) as the gradient of a scalar is a covariant vector.  The covariant formulation of the Fokker-Planck equation~\cite{Risken1989,Graham1977} will be useful when we consider more general coordinate transformations~\cite{Grabert1980,Fisenko2004}.

The nucleation flux in Eq.~(\ref{eq:15z}) is now transformed into
\begin{equation}
{\bm J}_{\bm \zeta}=-f_{\rm eq}\left({\bm n}\right){\bm V}^{\rm T}\left(\omega\right){\bm R}{\bm V}\left(\omega\right){\bm\nabla}_{\bm \zeta}\Phi
\label{eq:40z}
\end{equation}
at the saddle point.

By calculating the matrix products ${\bm V}^{\rm T}{\bm R}{\bm V}$, Eq.~(\ref{eq:40z}) is written explicitly by
\begin{eqnarray}
J_{\zeta_1} &=& -f_{\rm eq}\left({\bm n}\right)\Sigma\left(\omega\right)\frac{\partial \Phi}{\partial \zeta_1}+f_{\rm eq}\left({\bm n}\right)\Xi\left(\omega\right)\frac{\partial \Phi}{\partial \zeta_2},
\label{eq:42z} \\
J_{\zeta_2} &=& f_{\rm eq}\left({\bm n}\right)\Xi\left(\omega\right)\frac{\partial \Phi}{\partial \zeta_1}-f_{\rm eq}\left({\bm n}\right)\Upsilon\left(\omega\right)\frac{\partial \Phi}{\partial \zeta_2},
\label{eq:43z}
\end{eqnarray}
where
\begin{eqnarray}
\Sigma\left(\omega\right) &=&
R_{11}\cos^{2}\omega+R_{22}\sin^{2}\omega+2R_{12}\cos\omega\sin\omega,
\nonumber \\
\label{eq:44z} \\
\Xi\left(\omega\right) &=&
\left(R_{11}-R_{22}\right)\cos\omega\sin\omega+R_{12}\left(\sin^{2}\omega-\cos^{2}\omega\right),
\nonumber \\
\label{eq:45z} \\
\Upsilon\left(\omega\right) &=&
R_{11}\sin^{2}\omega + R_{22}\cos^{2}\omega - 2R_{12}\cos\omega\sin\omega
\nonumber \\
\label{eq:47z}
\end{eqnarray}
are the matrix elements of ${\bm V}^{\rm T}{\bm R}{\bm V}$.

We then choose the new coordinate $\zeta$ such that $\partial_{\zeta_2}\Phi=0$.  After manipulating Eqs.~(\ref{eq:42z}) and (\ref{eq:43z}) using the same procedure originally developed by Langer~\cite{Langer1969}, which is briefly summarized in Appendix A, we find
\begin{equation}
\tan\omega = s\pm\sqrt{s^2+r},
\label{eq:59z}
\end{equation}
where
\begin{eqnarray}
s &=& \frac{R_{22}G_{22}-R_{11}G_{11}}{2\left(R_{12}G_{11}+R_{22}G_{12}\right)},
\label{eq:60z} \\
r &=& \frac{R_{11}G_{12}+R_{21}G_{22}}{R_{12}G_{11}+R_{22}G_{12}},
\label{eq:61z}
\end{eqnarray}
are slightly different in definition from those used by others~\cite{Stauffer1976,Wilemski1999} since we have off-diagonal elements $R_{12}=R_{21}$.

Eq.~(\ref{eq:59z}) reduces to Eq.~(\ref{eq:32z})  that determines the steepest-descent direction of free-energy surface when $R_{11}=R_{22}$ and $R_{12}=R_{21}=0$.  Again, $+$ sign must be chosen when $R_{12}G_{11}+R_{22}G_{12}<0$ otherwise $-$ sign must be chosen.  Equation (\ref{eq:59z}) with $+$ sign also reduces to the well-known result~\cite{Stauffer1976,Wilemski1999} when $R_{12}=R_{21}=0$.  The $+$ sign was chosen by previous authors because they~\cite{Stauffer1976,Wilemski1999} considered the case when $G_{12}<0$.

Equation (\ref{eq:50z}) can now be written as
\begin{equation}
\frac{d^{2}\Phi}{d\zeta_{1}^{2}}=-\zeta_{1}\lambda\frac{d\Phi}{d\zeta_{1}}
\label{eq:62z}
\end{equation}
with
\begin{equation}
\lambda = L\left(\omega\right)/\Sigma\left(\omega\right).
\label{eq:63z}
\end{equation}
Since the boundary condition is given by
\begin{eqnarray}
\Phi\left(\zeta_1\right)=1,\;\;\;&\mbox{as}&\;\;\;\zeta_1\rightarrow -\infty,
\label{eq:64z} \\
\Phi\left(\zeta_1\right)=0,\;\;\;&\mbox{as}&\;\;\;\zeta_1\rightarrow \infty,
\label{eq:65z}
\end{eqnarray}
the solution of Eq.~(\ref{eq:62z}) is given by
\begin{equation}
\Phi\left(\zeta_1\right) = \frac{1}{2}{\rm erfc}\left(\sqrt{\frac{\lambda}{2}}\zeta_{1}\right),
\label{eq:67z}
\end{equation}
where ${\rm erfc}$ is the complementary error function.  Therefore, the gradient of $\Phi$ is given by
\begin{equation}
{\bm \nabla}\Phi={\bm e}_{\zeta_1}\frac{d\Phi}{d\zeta_1}
\label{eq:68z}
\end{equation}
with
\begin{equation}
{\bm e}_{\rm \zeta_1}=\cos\omega{\bm e}_{n_1}+\sin\omega{\bm e}_{n_2}
\label{eq:69z}
\end{equation}
from Eq.~(\ref{eq:35z}) and the rotation angle $\omega$ is given by Eq.~(\ref{eq:59z}).  It is easily confirmed by direct calculation that the direction of ${\bm e}_{\zeta_1}$ given by Eqs.~(\ref{eq:59z})-(\ref{eq:61z}) is in fact the direction of the eigenvector of the matrix ${\bm G}{\bm R}$ (Appendix).

Equation (\ref{eq:67z}) indicates that the steady state cluster distribution  $f\left(n_1,n_2\right)$ is smaller than the equilibrium distribution $f_{\rm eq}\left(n_1,n_2\right)$ since $\Phi<1$ and $f=f_{\rm eq}/2$ at the saddle point.  Also,
$\Phi=f/f_{\rm eq}$ becomes a universal function along $\zeta_1$ axis.  Numerical results of vapor to liquid binary nucleation~\cite{Wyslouzil1999,Li2000} also suggest that whole contour of $\Phi=f/f_{\rm eq}$ in two-dimensional $\left(n_1,n_2\right)$ space can also be relatively insensitive to the materials considered.  Therefore the universality of Eq.~(\ref{eq:67z}) could be used as a guide to deduce the reaction coordinate on the free-energy surface and the generalized coordinates that correspond to $n_1$ and $n_2$ using various techniques of computer simulations~\cite{tenWolde1996,Pan2004,Desgranges2007,Maibaum2008}.

\subsection{\label{sec:3-3}Direction of nucleation flux $\bm J$}
Thirdly, we will derive the direction of the nucleation flux $\bm J$ following the work of Wilemski~\cite{Wilemski1999}.  Now, we rotate $\left(\Delta n_1,\Delta n_2\right)$ axis by $\phi$ and introduce a new coordinate $\left(\mu_1,\mu_2\right)$.
Then, the nucleation flux given by Eq.~(\ref{eq:15z}) will be transformed into
\begin{equation}
{\bm J}_{\bm \mu}=f_{\rm eq}\left({\bm n}\right){\bm V}^{\rm T}\left(\phi\right){\bm R}{\bm V}\left(\phi\right){\bm\nabla}_{\bm \mu}\Phi
\label{eq:73z}
\end{equation}
similar to Eq.~(\ref{eq:40z}).

Using a procedure similar to that used in the previous subsection and Appendix A~\cite{Langer1969,Wilemski1999}, we obtain the relationship between $\phi$ and $\omega$ given by
\begin{equation}
\tan\phi=\frac{R_{12}+R_{22}\tan\omega}{R_{11}+R_{12}\tan\omega},
\label{eq:84z}
\end{equation}
which reduces to the formula derived previously~\cite{Stauffer1976,Wilemski1999} when $R_{12}=R_{21}=0$.

Similarly, the nucleation flux $\vec{J}$ is given by
\begin{equation}
\vec{J}=J_{\mu_1}{\bm e}_{\mu_1}
\label{eq:89z}
\end{equation}
with
\begin{equation}
J_{\mu_1}=f_{\rm eq}\left({\bm n}\right)\left({\rm det}{\bm R}\right)\left[\sigma\left(\phi\right)\right]^{-1}
\sqrt{\frac{\lambda}{2\pi}}\exp\left(-\frac{1}{2}\lambda\zeta_{1}^{2}\right),
\label{eq:90z}
\end{equation}
where $\sigma\left(\phi\right)$ is given by
\begin{eqnarray}
\sigma\left(\phi\right) &=&
\left[\left(R_{11}^{2}\sin^{2}\phi+R_{22}^{2}\cos^{2}\phi\right)\right.
\nonumber \\
&&\left.-2R_{12}\left(R_{11}+R_{22}\right)\cos\phi\sin\phi
+R_{12}^{2}\right]^{1/2}.
\label{eq:87z}
\end{eqnarray}
and its direction is
\begin{equation}
{\bm e}_{\mu_1}=\cos\phi{\bm e}_{n_1}+\sin\phi{\bm e}_{n_2}
\label{eq:91z}
\end{equation}
with the rotation angle $\phi$ given by Eq.~(\ref{eq:84z}).

Equation (\ref{eq:84z}) tells us that the direction ${\bm v}_{\mu_1}^{\rm T}=\left(\cos\phi,\sin\phi\right)$  of the nucleation flux ${\bm e}_{\mu_1} (\propto{\bm e}_{\xi_1})$ and the direction ${\bm v}_{\zeta_1}^{\rm T}=\left(\cos\omega,\sin\omega\right)$ of the gradient of $\Phi$ (${\bm e}_{\zeta_1}$) is related through ${\bm v}_{\mu_1}\propto {\bm v}_{\xi_1}\propto {\bm R}{\bm v}_{\zeta_1}$, which again confirms the statement in the Appendix that the direction of the eigenvector of the negative eigenvalue $\lambda_1$ of the matrix ${\bm G}{\bm R}$ represents the direction of the gradient of $\Phi$ and that of the matrix ${\bm R}{\bm G}$ represents the direction of the nucleation flux $\bm J$ at the saddle point.

\subsection{\label{sec:sec3-4}Nucleation rate}
Finally, we will consider the nucleation rate.  Since the general theory was already given by Trinkaus~\cite{Trinkaus1983}, we will simply apply his theory to the two-dimensional case in this subsection.  In Appendix B, we give a detailed derivation of the formula for the nucleation rate by Trinkaus~\cite{Trinkaus1983} and the relationship between the Trinkaus' theory and Kramers-Langer-Berezhkovskii-Szabo (KLBS) theory~\cite{Kramers1940,Langer1969,Berezhkovskii2005}.

To simplify the notation, we will introduce matrices
\begin{equation}
{\bm r}={\bm R}^{1/2}=
\begin{pmatrix}
r_{11} & r_{12} \\
r_{21} & r_{22}
\end{pmatrix}
\label{eq:92z}
\end{equation}
and
\begin{equation}
{\bm \Gamma}={\bm r}{\bm G}{\bm r}
=
\begin{pmatrix}
\Gamma_{11} & \Gamma_{12} \\
\Gamma_{21} & \Gamma_{22}
\end{pmatrix}
\label{eq:93z}
\end{equation}
defined in Eq.~(\ref{eq:a20}).  We also introduce the rotation of coordinate ${\bm V}\left(\alpha\right)$ in Eq.~(\ref{eq:17z}) with rotation angle $\alpha$ that corresponds to the orthonormal transformation in Eq.~(\ref{eq:a27}).

Now, the nucleation flux is given as Eqs.~(\ref{eq:89z})-(\ref{eq:91z}) is also written by Eqs.~(\ref{eq:a40})  and (\ref{eq:a41}).  Therefore
\begin{equation}
\lambda\zeta_{1}^{2}=-\beta\lambda_1\xi_{1}^{2}.
\label{eq:109z}
\end{equation}
and
\begin{equation}
\lambda=\frac{\beta\lambda_{1}}{{\rm det}{\bm R}}\left(\frac{\sigma\left(\phi\right)}{\rho\left(\phi\right)}\right)^{2}.
\label{eq:111z}
\end{equation}
since
\begin{equation}
\left|{\bm e}_{\xi_1}\right|={\rm det}{\bm R}^{1/2}/\rho\left(\phi\right)
\label{eq:106z}
\end{equation}
and
\begin{eqnarray}
\rho\left(\phi\right) &=&
\left[\left(r_{11}^{2}\sin^{2}\phi+r_{22}^{2}\cos^{2}\phi\right)\right.
\nonumber \\
&&\left.-2r_{12}\left(r_{11}+r_{22}\right)\cos\phi\sin\phi+r_{12}^{2}\right]^{1/2},
\label{eq:104z}
\end{eqnarray}
using the direction  $\phi$ of the nucleation flux.

Using the same procedure as that used in section \ref{sec:3-1}, we have one negative ($\lambda_1$) and one positive ($\lambda_2$) eigenvalues
\begin{eqnarray}
\lambda_1 &=& \left(\left(\Gamma_{11}+\Gamma_{22}\right)-\sqrt{\left(\Gamma_{11}
+\Gamma_{22}\right)^2-4{\rm det}{\bm\Gamma}}\right)/2<0, \nonumber \\
\label{eq:97z} \\
\lambda_2 &=& \left(\left(\Gamma_{11}+\Gamma_{22}\right)+\sqrt{\left(\Gamma_{11}
+\Gamma_{22}\right)^2-4{\rm det}{\bm\Gamma}}\right)/2>0, \nonumber \\
\label{eq:98z}
\end{eqnarray}
and the rotation angle $\alpha$ is given by
\begin{equation}
\tan\alpha=\frac{\Gamma_{22}-\Gamma_{11}}{2\Gamma_{12}}
\pm\sqrt{\left(\frac{\Gamma_{22}-\Gamma_{11}}{2\Gamma_{12}}\right)^{2}+1}
\label{eq:99z}
\end{equation}
where $+$ sign must be chosen if $\Gamma_{12}<0$ otherwise $-$ sign must be chosen.
Since, the direction of nucleation flux ${\bm e}_{\xi_1}$ in Eq.~(\ref{eq:a41}) must be parallel to ${\bm e}_{\mu_1}$ in Eq.~(\ref{eq:91z}), we have
\begin{equation}
\tan\phi=\frac{r_{12}+r_{22}\tan\alpha}{r_{11}+r_{12}\tan\alpha}
\label{eq:101z}
\end{equation}
which will be reduced to the form obtained by Wilemski~\cite{Wilemski1999} as $r_{11}=R_{11}^{1/2}$ and $r_{22}=R_{22}^{1/2}$ when $R_{12}=R_{21}=0$ ($r_{12}=r_{21}=0$).

Finally, the nucleation rate is given by
\begin{equation}
I=f_0 e^{-\beta G^{*}}\sqrt{{\rm det}{\bm R}\frac{\left|\lambda_{1}\right|}{\lambda_2}},
\label{eq:113z}
\end{equation}
from Eq.~(\ref{eq:a48}).

In the next section, we will study the three angles $\theta$, $\omega$ and $\phi$ which represent the steepest-descent direction on the free-energy surface, gradient of $\Phi$ (commitment probability), and the nucleation flux $\bm J$.  These directions are characterized by $\theta$, $\omega$ and $\phi$ which correspond to the directions of the eigenvector of the negative eigenvalue of the matrix $\bm G$, ${\bm G}{\bm R}$ and ${\bm R}{\bm G}$, respectively~\cite{Langer1969,Berezhkovskii2005}.

\section{\label{sec:sec4}Application to the composite nucleus}

In this section, we will apply the theory of nucleation flux of binary nucleation at the saddle point developed in sec. \ref{sec:sec3} to the problem of nucleation flux of composite nucleus considered in sec. \ref{sec:sec2}.  In this case, the reaction rate matrix is non-diagonal,
\begin{equation}
{\bm R}=
\begin{pmatrix}
\kappa^{+} & -\kappa^{+} \\
-\kappa^{+} & \kappa^{+}+\alpha^{+}
\end{pmatrix}
\label{eq:114z}
\end{equation}
as given in Eq.~(\ref{eq:15z}).  Similar models with non-diagonal reaction matrix were considered by Russel~\cite{Russell1968} as a linked flux model and by Djikaev~\cite{Djikaev2002} to study the nucleation in deliquescence using the theory of Melikhov et al.~\cite{Melikhov1991}.  However the latter author used the theory~\cite{Melikhov1991} which reduces the two-dimensional problem to an approximate one-dimensional problem.  Furthermore,  only a numerical result of deliquescence path is given  and a detailed theoretical analysis is missing~\cite{Djikaev2002,Russell1968}.  Here, we will study the two-dimensional problem directly using rigorously formulated theory without resorting to an approximation.

Since, the eigenvalues of $\bm \Gamma$ are the same as those of the product ${\bm R}{\bm G}$ and ${\bm G}{\bm R}$, they can easily be obtain from Eqs.~(\ref{eq:97z}) and (\ref{eq:98z}) explicitly as
\begin{eqnarray}
\lambda_{1}&&=\frac{1}{2}\left[\left(\alpha^{+} G_{22}+\kappa^{+}\left(G_{11}+G_{22}-2G_{12}\right)\right) \right. \nonumber \\
&& -\left.\sqrt{\left(\alpha^{+} G_{22}+\kappa^{+}\left(G_{11}+G_{22}-2G_{12}\right)\right)^{2}-4\alpha^{+} k^{+}{\rm det}{\bm G}}\right], \nonumber \\
\label{eq:115z} \\
\lambda_{2}&&=\frac{1}{2}\left[\left(\alpha^{+} G_{22}+\kappa^{+}\left(G_{11}+G_{22}-2G_{12}\right)\right) \right. \nonumber \\
&& +\left.\sqrt{\left(\alpha^{+} G_{22}+\kappa^{+}\left(G_{11}+G_{22}-2G_{12}\right)\right)^{2}-4\alpha^{+} k^{+}{\rm det}{\bm G}}\right], \nonumber \\
\label{eq:116z}
\end{eqnarray}
where $\lambda_1<0$ and $\lambda_2>0$ since ${\rm det}{\bm G}<0$ at the saddle point.  Using these eigenvalues $\lambda_1$ and $\lambda_2$, the nucleation rate is given by
\begin{equation}
I=f_0 e^{-\beta G^{*}}\sqrt{\alpha^{+} \kappa^{+}\frac{\left|\lambda_{1}\right|}{\lambda_2}},
\label{eq:117z}
\end{equation}
from Eq.~(\ref{eq:113z}) since ${\rm det}{\bm R}=\alpha^{+}\kappa^{+}$.  The nucleation rate depends not only on the rate $\alpha^{+}$ from the metastable parent phase to the metastable intermediate phase but also on the rate $\kappa^{+}$ from the metastable intermediate phase to the stable new phase.  Therefore, even when $\alpha^{+}\gg \kappa^{+}$, the nucleation problem does not reduce to one-component unary problem~\cite{Trinkaus1983,Melikhov1991,Shi1990,Zitserman1990}.  Since there is only one saddle point, a formula for the successive nucleation rate~\cite{Iwamatsu2011,Valencia2004,Valencia2006} cannot apply.

The steepest-descent direction $\theta$ of free-energy surface is given by Eq.~(\ref{eq:32z}), which is determined solely by the free-energy surface near the critical point given by Eq.~(\ref{eq:25z}).  The direction $\omega$ of the gradient of $\Phi$, however, is given by a more complex formula in Eq.~(\ref{eq:59z}) because the effect of the anisotropy of reaction rate cannot be ignored.  Using the explicit form of reaction rate in Eq.~(\ref{eq:114z}) we have the direction $\phi$
\begin{equation}
\tan\phi=\frac{-\kappa^{+}+\left(\kappa^{+}+\alpha^{+}\right)\tan\omega}{\kappa^{+}-\kappa^{+}\tan\omega},
\label{eq:120z}
\end{equation}
once we know $\omega$ from Eqs.~(\ref{eq:59z}).

\begin{figure}
\includegraphics[width=1.0\linewidth]{fig2.EPS
\caption{Relation between $\omega$ and $\phi$ for various relative reaction rate $\tilde{\alpha}=\alpha^{+}/\kappa^{+}$.  The angle $\phi$ is an increasing function of $\omega$ starting from $\phi=-45^{\circ}$ at $\omega=0^{\circ}$.  The straight line indicates the line when $\phi=\omega$.  Below this line, the direction of the nucleation current is deflected from the direction of ${\rm grad}\Phi$ to the right.  }
\label{fig:2x}
\end{figure}

Figure 2 shows the relationship between $\omega$ and $\phi$ for various relative reaction rate $\tilde{\alpha}=\alpha^{+}/\kappa^{+}$.  The angle $\phi$ is an increasing function of the angle $\omega$ starting from $\phi=-45^{\circ}$ at $\omega=0^{\circ}$ from Eq,~(\ref{eq:120z}).  The direction $\phi$ of nucleation current $\bm J$ changes from right ($\phi<\omega$) to left ($\phi>\omega$) of the direction $\omega$ of the gradient of $\Phi$ as the angle $\omega$ is increased.  As the angle $\omega$ is further increased, the angle $\phi$ becomes $\phi=90^{\circ}$ when $\omega=45^{\circ}$ from Eq.~(\ref{eq:120z}) and continue to increase.  Therefore, as the direction $\omega$ of the gradient $\Phi$ increases, the nucleation flux at the saddle point turns toward the $n_2$-axis and the component $n_2$ of the intermediate metastable phase rather than $n_1$ of the stable new phase tends to grow.

When $\tilde{\alpha}<1$ ($\alpha^{+}<\kappa^{+}$), the incoming rate $\alpha^{+}$ from the metastable parent phase to the intermediate metastable phase ($n_2$) is slower than the internal transformation rate $\kappa^{+}$ from the intermediate metastable phase to the new stable phase.  Then the stable new phase ($n_1$) tends to grow faster than the intermediate metastable phase ($n_2$), and the direction $\phi$ of nucleation flux remains toward the direction of $n_1$ axis and the inequality $\phi<\omega$ follows as long as $\omega$ is not as large as $45^{\circ}$ (see Fig.~\ref{fig:2x}, $\tilde{\alpha}=0.1, 0.7$).  When  $\tilde{\alpha}>1$, the incoming rate $\alpha^{+}$ is faster than the internal transformation rate $\kappa^{+}$ and the intermediate metastable phase grows faster than the new stable phase.  Then the direction $\phi$ soon becomes larger than $\omega$ as $\omega$ is increased from 0  (see Fig.~\ref{fig:2x}, $\tilde{\alpha}=2.0, 5.0$).

In order to study the relationship and the absolute magnitude of various angles $\theta$, $\omega$, and $\phi$, we have to specify the free-energy surface $G\left({\bm n}\right)$.  To enable the qualitative idea of the composite nucleus, we will give an example to illustrate the concept and study a model system specified by the free-energy landscape
\begin{eqnarray}
G\left({\bm n}\right)&=&G^{*} \nonumber \\
&+&\frac{1}{2}G_{22}\left[\left(1+\frac{2}{\sqrt{3}}{\rm g}_{12}\right)\Delta n_{1}^{2}+\Delta n_{2}^2+2{\rm g}_{12}\Delta n_1\Delta n_2\right] \nonumber \\
\label{eq:121z}
\end{eqnarray}
with ${\rm g}_{12}<0$.  In this case $G_{11}=G_{22}\left(1+2/\sqrt{3}{\rm g}_{12}\right)$, $G_{12}=G_{22}{\rm g}_{12}$, and we find from Eq.~(\ref{eq:32z}):
\begin{equation}
\tan\theta=\frac{1}{\sqrt{3}},\;\;\;\mbox{or}\;\;\;\theta=30^{\circ}.
\label{eq:122z}
\end{equation}
In this model, only ${\rm g}_{12}<0$ is the parameter that characterizes the free-energy landscape.  In order to make the origin $(\Delta n_1, \Delta n_2)=(0,0)$ the saddle point, ${\rm det}{\bm G}$ must be negative and  ${\rm g}_{12}$ must satisfy ${\rm g}_{12}<-1/\sqrt{3}=-0.577$.

\begin{figure}[htbp]
\begin{center}
\includegraphics[width=0.75\linewidth]{fig3z.eps}
\end{center}
\caption{
An example of the model free-energy landscape of composite nucleus when $\theta=30^{\circ}$ defined by Eq.~(\ref{eq:121z}) when ${\rm g}_{12}=-0.7$.  The origin is the saddle point shifted to $(0, 0)$. The steepest-descent direction is always fixed to $\theta=30^{\circ}$ indicated by a pink arrow from the saddle point indicated by a pink point. The direction $\omega$ of the gradient of $\Phi$ in Fig.~\ref{fig:4} is indicated by a red thin ($\tilde{\alpha}=0.0$) and a thick ($\tilde{\alpha}=7.0$) arrows. The direction $\phi$ of the nucleation flux $\bm J$ is not shown as it stays near the steepest-descent direction $\theta=30^{\circ}$ represented by a pink arrow (see also Fig.~\ref{fig:5}).  }
\label{fig:3}
\end{figure}

In Figs.~\ref{fig:3} and \ref{fig:4}, we show contour plots of typical free-energy landscape of our model in Eq.~(\ref{eq:121z}) when ${\rm g}_{12}=-0.7$ and ${\rm g}_{12}=-5.0$.  The saddle point is located at the origin $(0, 0)$.   Although the direction of the steepest-descent is fixed to $30^{\circ}$, the saddle point and the valley that lead to the saddle becomes wider and shallower as the parameter ${\rm g}_{12}$ becomes more negative (cf. Figs.~\ref{fig:3} and \ref{fig:4}).

\begin{figure}[htbp]
\begin{center}
\includegraphics[width=0.75\linewidth]{fig4z.eps}
\end{center}
\caption{
The same as Fig.~\ref{fig:3} when ${\rm g}_{12}=-5.0$.  Now the value of ${\rm g}_{12}$ becomes more negative than that in Fig.~\ref{fig:3} and the saddle point and the valley that leads to the saddle become wider. The direction $\omega$ of the gradient of $\Phi$ in Fig.~\ref{fig:5} is indicated by green thin ($\tilde{\alpha}=0.0$) and thick ($\tilde{\alpha}=7.0$) arrows of solid line. The direction $\phi$ of nucleation flux $\bm J$ is indicated by green thin ($\tilde{\alpha}=0.0$) and thick ($\tilde{\alpha}=7.0$) arrows of broken line.  }
\label{fig:4}
\end{figure}

Figure~\ref{fig:5} shows the angles  $\omega$ and $\phi$ as the functions of the relative reaction rate $\tilde{\alpha}$.  Since we always have $R_{12}G_{11}+R_{22}G_{12}=-\kappa^{+}G_{11}+\left(\kappa^{+}+\alpha^{+}\right)G_{12}<0$ because $G_{11}>0$ and $G_{12}<0$, we took $+$ sign in Eq.~(\ref{eq:59z}).   We can easily show from Eqs.~(\ref{eq:59z}),  (\ref{eq:120z}) that
\begin{equation}
\tan\omega\rightarrow 1-\frac{G_{22}-G_{12}}{G_{11}+G_{22}-2G_{12}}\left(\frac{\alpha^{+}}{\kappa^{+}}\right),\;\;\;\mbox{as}\;\;\;\alpha^{+}\rightarrow 0^{+}.
\label{eq:123z}
\end{equation}
Therefore
\begin{equation}
\omega\rightarrow 45^{\circ},\;\;\;\mbox{as}\;\;\;\alpha^{+}\rightarrow 0^{+}
\label{eq:124z}
\end{equation}
and
\begin{equation}
\tan\phi\rightarrow \frac{G_{11}-G_{12}}{G_{22}-G_{12}},\;\;\;\mbox{as}\;\;\;\alpha^{+}\rightarrow 0^{+}
\label{eq:125z}
\end{equation}
from Eq.~(\ref{eq:120z}).  The angle $\omega$ that specifies the direction of the gradient of $\Phi$ always starts from $\omega=45^{\circ}$ at $\tilde{\alpha}=0$ irrespective of the shape of the free-energy surface $G\left({\bm n}\right)$.

\begin{figure}
\includegraphics[width=1.0\linewidth]{fig5z.eps
\caption{The angle $\omega$ and $\phi$ as the functions of the relative reaction rate $\tilde{\alpha}$ for various free-energy parameters ${\rm g}_{12}$ when $\theta=30^{\circ}$. }
\label{fig:5}
\end{figure}

As the relative reaction rate $\tilde{\alpha}$ is increased, the incoming flux from the metastable parent phase to the surrounding wetting layer of the intermediate metastable phase increases.  Then the intermediate metastable component $n_2$ of the composite nucleus increases and the nucleation flux tends toward direction of $n_2$ axis and the angle $\phi$ increases.  In Fig.~\ref{fig:5}, the angle $\omega$, which is the direction of gradient of $\Phi$, starts from $\omega=45^{\circ}$ and rotates clockwise towards $n_{1}$ axis, while the angle $\phi$, which is the direction of nucleation flux, starts from the right-hand side of the direction $\theta$ and rotates anti-clockwise and turns to the left-hand side of the direction of $\theta$ as the relative reaction rate $\tilde{\alpha}$ is increased.

In Figs.~\ref{fig:3} and \ref{fig:4}, we show the direction of $\omega$ and $\phi$ at $\tilde{\alpha}=0.0$ and $7.0$ by thin and thick arrows using the same color as those used in Fig.~\ref{fig:5}. In Fig.~\ref{fig:4}, the arrow of solid line which represents the direction of $\omega$ of the gradient of $\Phi$ rotates clockwise, while that of broken line which represents the direction of $\phi$ of the nucleation flux $\bm J$ rotates anti-clockwise as $\tilde{\alpha}$ is increased.  It can be shown analytically that both the directions $\omega$ and $\phi$ become the same as the steepest-descent direction $\theta=\phi=\omega=30^{\circ}$ at $\tilde{\alpha}=2/\sqrt{3}$ (see Fig.~\ref{fig:5}).  When ${\rm g}_{12}=-1/\sqrt{3}$, the direction $\phi$ is fixed to the steepest-descent direction $\phi=\theta=30^{\circ}$ because the free-energy landscape becomes the corridor of valley rather than the saddle.

In Fig.~\ref{fig:7}, we show the direction $\omega$ and $\phi$ as the function of the relative reaction rate $\tilde{\alpha}$ when the steepest-descent direction is $\theta=60^{\circ}$ using the model free-energy landscape
\begin{eqnarray}
G\left({\bm n}\right)&=&G^{*} \nonumber \\
&+&\frac{1}{2}G_{22}\left[\left(1-\frac{2}{\sqrt{3}}{\rm g}_{12}\right)\Delta n_{1}^{2}+\Delta n_{2}^2+2{\rm g}_{12}\Delta n_1\Delta n_2\right]. \nonumber \\
\label{eq:126z}
\end{eqnarray}
As the relative reaction rate $\tilde{\alpha}$ is increased, the angle $\omega$ starts from $\omega=45^{\circ}$ and rotate clockwise towards $n_{1}$ axis again.  The angle $\omega$ is relatively insensitive to the parameter ${\rm g}_{12}$ of the free-energy surface not only in Fig.~\ref{fig:7} but also in Fig~\ref{fig:5}.  Several authors observed similar behaviors of $\omega$ and $\Phi$ in vapor-liquid nucleation~\cite{Wilemski1999,Li1999,Wyslouzil1999} which are relatively insensitive to the system considered.  In our case, the decreasing $\omega$ indicates that the size distribution function $f$ will be characterized mainly by the number of molecules $n_{1}$ in the final stable phase ($f\left(n_1, n_2\right)\simeq f\left(n_1\right)$) as $\tilde{\alpha}$ is increased.  The size distribution $f\left(n_1,n_2\right)$ is flat along $n_2$ axis for fixed $n_1$ and many clusters with the same $n_1$ and different $n_2$ coexist.

On the other hand, the angle $\phi$ starts from the angle $\phi>60^{\circ}$ calculated from Eq.~(\ref{eq:125z}) that is larger than the steepest-descent direction $\theta=60^{\circ}$ and continues to increase.   When ${\rm g}_{12}=-\sqrt{3}$, the direction $\phi$ is fixed again to the steepest-descent direction $\phi=\theta=60^{\circ}$ because the free-energy landscape becomes corridor of a valley.  In Fig.~\ref{fig:7} the angle $\omega$ and $\phi$ will never coincide.  As expected, the direction  $\phi$ of the nucleation flux is closer than the direction $\omega$ of the gradient of $\Phi$ to the direction $\theta$ of the steepest-descent of the free-energy surface.  Our results in Figs.~\ref{fig:5} and \ref{fig:7} clearly show that the difference between the steepest-descent direction $\theta$ and the nucleation flux direction $\phi$ depends not only on the difference of the reaction rates represented by $\tilde{\alpha}$ but also to the shape of the free-energy surface as pointed out by Wyslouzil and Wilemski~\cite{Wyslouzil1995}.

\begin{figure}[htbp]
\includegraphics[width=1.0\linewidth]{fig7.EPS}
\caption{
The same as Fig~\ref{fig:5} when  $\theta=60^{\circ}$. }
\label{fig:7}
\end{figure}

The deviation of the direction of the nucleation flux from that of the steepest-descent direction of free-energy surface has been pointed out repeatedly in different context from condensed matter~\cite{Russell1968,Trinkaus1983, Temkin1984,Greer1990} to liquid-vapor systems~\cite{Reiss1950,Stauffer1976,Wilemski1999}.  However those authors except Trinkaus~\cite{Trinkaus1983} used various approximations, in particular, the reaction rate matrix is assumed to be diagonal.  Therefore, their conclusion remained qualitative.  In contrast, we have used the theory~\cite{Kramers1940,Langer1969,Berezhkovskii2005} which includes the non-diagonal element of the reaction rate matrix $\bm R$ to study the direction of nucleation flux.  Therefore, our result will be useful to study the nucleation flux of more complex situation and will contribute to deeper understanding of a more complex nucleation process in the future.

\section{\label{sec:sec5}Concluding remark}
In this paper, we have studied the nucleation flux of composite nucleus that consists of a stable new phase surrounded by a wetting layer of intermediate metastable phase nucleated from the metastable parents phase.  By using the results of exact formulation of nucleation flux~\cite{Kramers1940,Langer1969,Berezhkovskii2005}, we have shown that the nucleation rate is determined not only from the reaction rate from the intermediate metastable phase to the stable new phase but also from that from the metastable parent phase to the intermediate metastable phase.  We have also shown that the direction of nucleation flux deviates significantly from that of the steepest-descent direction of the free-energy surface.

Kinetics of composite nucleus has been considered by several authors in different contexts.  Kelton and coworkers~\cite{Kelton2000, Diao2008} have extended the theory of linked flux theory of nucleation by Russell~\cite{Russell1968} and studied the partitioning transition where the interplay of diffusional and interfacial processes are linked.  In this case, the nucleus consists of a core of new stable phase surrounded by a layer of pre-nucleus zone.  They~\cite{Kelton2000,Diao2008} did not show any analysis of the nucleation flux since they do not appear to be interested in the size distribution of nucleus which is directly comparable to a real experiment.  In fact, Kelton~\cite{Kelton2003} applied the theory to the oxygen precipitation in silicon and obtained a precipitation density that agrees fairly well with their experiment.  Russell~\cite{Russell1968} showed only numerically that the nucleation flux will deviate from the steepest-descent direction of free-energy surface. Very recent study by Peters~\cite{Peters2011}, however, does not use the wetting layer as a model of diffusion but considers the coupling of diffusion and nucleation directly.  His results clearly demonstrated that the nucleation pathway is influenced by the diffusion represented by along-wavelength fluctuation and also by the initial state and questioned the adequacy of the composite nucleus model. Recent studies~\cite{Philippe2011,Lutsko2012} using the dynamical density functional theory also pointed out the importance of long-wavelength fluctuation as the pre-nucleus state and questioned the composite nucleus model as a model of two-step nucleation.

A similar composite nucleus that appears in the problem of deliquescence has been considered by Djikaev~\cite{Djikaev2002}.  In this model~\cite{Djikaev2002} the wetting layer is not merely a dense pre-nucleus zone~\cite{Kelton2000,Diao2008,Russell1968} but is rather a true thermodynamic liquid phase.  Therefore his model is closer to ours~\cite{Iwamatsu2011} since we consider the wetting layer as a metastable thermodynamic phase.  For a realistic model, he~\cite{Djikaev2002} showed only numerically that the deliquescence path, which is the direction of nucleation flux, deviates from the direction of the equilibrium path of the steepest-descent on the free-energy surface.  In contrast, our analytical analysis is not restricted to specific materials and will be useful for future extension and applications to various complex materials.

It must be noted, however, our analysis is completely confined to the steady-state process.  Therefore, transient properties~\cite{Kelton2000,Diao2008,Shi1990,Zitserman1990} and the problem of time scale~\cite{Melikhov1991,Djikaev2002} are out of scope for this study, which can only be examined numerically by solving coupled Master equation~\cite{Kelton2000,Diao2008,Wyslouzil1996}.  Also, our analysis assumed that the nucleation flux goes through the saddle point.  Saddle point avoidance~\cite{Trinkaus1983,Zitserman1990,Berezhkovskii1995,Wyslouzil1995,Wilemski1995} will be important if the anisotropy of the reaction matrix $R$ is large or the ridge between saddle point is low, which can occur at high temperatures or near the spinodal point.  In such a case, ridge-crossing rather than saddle-crossing may occur.  Then the nucleation flux will spread over the whole phase space and the picture used in this study may break down.  In such a case, the growth of composite nucleus will be expected to be more complex.

Finally, our analysis will be useful in the study of nucleation by attachment and detachment of molecules as well as the study of various barrier crossing phenomena such as chemical reactions and conformation transformation of molecules.  Newly developed various computer simulation techniques make it possible not only to explore the free-energy landscape~\cite{tenWolde1996,Desgranges2007,Maibaum2008} but also to extracts the reaction rate matrix~\cite{Ma2006,Peters2009} and identify reaction coordinate~\cite{Ma2005,Peters2006,Peters2007,Bolhuis2009,Peters2009}.  Therefore, our analysis will be useful to deduce important parameters from those numerical data extracted from simulations.

Our analysis will also be useful even when multiple saddles exist~\cite{Iwamatsu2011,Ray1985,Wagner2001,Chen2003,Nellas2006} and multiple pathways of nucleation coexist.  In this case, the nucleation flux may fork into multiple streams and an analogy to the electric current in a circuit with parallel and series connections of conductors or resistors will be useful~\cite{Iwamatsu2012} as long as the fluxes go through those saddles.  Our results will be valid if some of those multiple saddles correspond to the composite nucleus.

\begin{acknowledgments}
This work was supported by the Grant-in-Aid for Scientific Research [Contract No.(C)22540422] from Japan Society for the Promotion of Science (JSPS) and a project for strategic advancement of research infrastructure for private universities, 2009-2013, from MEXT, Japan.  The author is grateful to one of the reviewers who pointed out importance of KLBS theory for his constructive suggestion to improve the original manuscript.

\end{acknowledgments}

redpajama_set_name: RedPajamaArXiv


Welp. I was really excited about the Grafmatic backs. Not so much now.
If you don't shoot 4×5, you don't really understand the frustration of shooting with the plastic cassettes. They work well, but they only fit 2 shots per cassette. Which isn't much.
The Grafmatic will hold 6! Whoa momma! And it's the size of about 2 plastic holders and weighs about as much. And cycling through each photo is a piece of pie. No need to flip over the cassette etc.
They were a dream come true. Unfortunately the dream abruptly ended as I developed my first photos.
Back to the drawing board and back to the cassettes!
I think I have it figured out! I've simply been doing it wrong. I'll have to post on why that is but after reading this thread, I'm confident the light leakiness is my fault.


text:
Debunking The Myths Around The 6 Scariest Animals In Australia
Watching a crocodile feeding at Hartley's Crocodile Adventures in Queensland
Australia is a fascinating island country at the bottom of the globe, the only inhabited continent located entirely south of the equator. While its intrigue captures the attention of travelers, the distance and time it takes to get there is one thing that can be off-putting to many.
Buckle up for a 15-hour flight from Los Angeles to Sydney, or be prepared for over 21 hours of flying time from London. Then, just as you start recovering from jetlag, you have to fend off all those scary creatures that roam the country; hell-bent on killing you! Or are they?
Let's take a look at the six scariest creatures to see if they are genuinely worthy of our fear.
The lagoon pool at Airlie Beach for safe swimming
Photo credit: Tourism Whitsundays
1. Box Jellyfish
When the award for the most venomous creature in the world is up for grabs, the box jellyfish had better have an acceptance speech written. This creature, also known as a sea wasp, can cause death in as little as 2–4 minutes.
The bell (head) of the jellyfish grows to a foot across and each corner has 15 tentacles that can stretch as long as 10 feet. It is from the tentacles that potent venom is delivered. The nervous system is affected within 4 minutes of the fatal sting and the toxin locks the heart into a contracted state. Death comes quickly.
But the good news is that only 79 people have been killed by box jellyfish since record-keeping began in 1879. So, the odds are stacked in your favor. And, they're only found in the warm waters of northern Australia, mainly from November–May. Draw a line halfway through the map of Australia and you should be fine to swim anywhere south of that line.
Pro Tip: The best defense is prevention. If visiting regions where they live, swim only within stinger enclosures. Always wear a stinger suit if you swim outside the enclosures. Locations such as Cairns and Airlie Beach in Queensland have protective lagoons with sandy beaches and grassy areas, so all family members can cool off safely away from ocean waters.
Not all Australian snakes are dangerous.
2. Taipan Snake
We can't deny that the world's most venomous snake, the taipan, is endemic to Australia. Their venom is so potent it can kill a human within hours if a sufficient amount is injected.
The average length of a taipan is 6–8 feet and they, too, are found in the warmer, wetter temperate-to-tropical coastal regions. Their habitat includes woodlands, grazing paddocks, hollow logs, piles of vegetation and litter, and sugar cane fields — anywhere there is an abundance of rats and mice.
Take comfort in knowing that rats, mice, and birds are their preferred diet. Humans aren't featured on their menu, but they will strike quickly and repeatedly if under a surprise siege.
Pro Tip: While known to be highly nervous and alert, taipans prefer a quick vanishing act over engaging in a confrontation. Official estimates declare there are about 3,000 all-species snake bite cases annually in Australia; 300–500 require anti-venom with an average of two fatalities a year.
The highly-feared great white shark
Photo credit: Australian Coastal Safaris
3. Great White Shark
We've all read Peter Benchley's novel Jaws, or seen Spielberg's movie adaptation. I was 14 when my fingernails dug deeply into the cinema seating on my one and only viewing of the movie. I screamed and looked away; the spine-chilling scenes of a great white shark attacking humans and boats were too much. That movie kept people out of the water for years. For many, the vision remains in the depths of our brain, parcelled as "things to be scared of."
These sharks inhabit Australian waters, but humans are also not on their usual menu. The chance of being a victim of a great white shark attack in Australia is well over 50 million to one.
Pro Tip: The International Shark Attack File lists the number of unprovoked and provoked attacks, with the latter almost doubled. Therein lies the first tip: as the saying goes, "don't poke the bear!" Avoiding unprovoked attacks includes not swimming at dusk or in darkness, not wearing sparkling objects and swimming only in clear waters. If the beach is patrolled, swim between the flags, and leave the water quickly if schools of fish gather.
A safer way to get close to crocodiles
4. Saltwater Crocodile
Steve Irwin, the late, great Aussie crocodile hunter, and the fictitious movie character Crocodile Dundee, gave the impression that crocodile wrestling is a daily pastime for Australians. The truth is that it is a big "nope!"
This fearsome animal is the largest living reptile in the world. In Australia, they can be found in coastal waters, estuaries, lakes, inland swamps, and marshes in the northern areas of Queensland, the Northern Territory, and upper regions of western Australia. Growing up to 7 meters in length (almost 23 feet) and living up to 80 years of age, the fascination around these creatures often counteracts the fear that surrounds them.
They are not fussy eaters; their diet consists mostly of birds and fish. Still, they'll take an unsuspecting kangaroo, water buffalo, or foolhardy human who enters their space. However, you can take comfort in that they're reported to kill less than one human annually on average.
Pro Tip: Don't seek them out in the wild. Never swim or get near the water's edge where signs indicate their presence. Several safe, up-close opportunities exist in which to view them. You can take a river cruise and watch in safety as wild crocodiles are fed. The Crocosaurus Cove in Darwin provides an intimate underwater viewing system where those game enough can enter the water and appreciate the size of these reptiles from the safety of a clear acrylic cylinder.
Sydney Funnel-web spiders burrow under rocks and logs in the forests of New South Wales.
5. Sydney Funnel Web Spider
As the name suggests, the Sydney Funnel-web spider is found only in New South Wales. This little guy grows to 1.5 inches and burrows into the ground, predominantly under logs and rocks in forested areas. They spin silk trip lines around their burrow and rush out when tasty beetles, cockroaches, small lizards, or snails herald their presence when brushing against the silk.
Contrary to popular belief, they don't jump onto or chase people. While they may stumble into a swimming pool or take cover in a shoe left outside, they don't live in houses. Although their fearsome reputation is often exaggerated, they deserve respect as their venom is the most toxic of any spider to humans and can result in death. The venom of the females and juvenile spiders is less harmful.
Pro Tip: First aid consisting of a pressure bandage and immobilization should be administered immediately after a bite is suspected. The victim should then be taken to a hospital where an anti-venom can be given. There have been no fatalities from Sydney Funnel-web spider bites since the anti-venom was developed in 1981.
Stonefish inhabit shallow estuaries and reefs.
6. Stonefish
Suppose you venture into shallow bays, estuaries, and reef flats in the coastal regions of northern Australia. In that case, you may encounter the estuarine stonefish. This experience won't be pleasant. For a start, the stonefish is just plain ugly. However, it does hold the award for the most venomous fish and deserves respect on that merit. But to look at it, it is junkyard ugly.
Around 1 foot in length, the skin of the stonefish is devoid of scales, covered in warts and algae. Its beady eyes protrude from a raised bony structure on a grooved and pitted head. Their grotesque head and warty body allow them to be masters at camouflaging well against the rocky sea bottom.
It is pretty rare to actually see the stonefish; however, should you unwittingly step near or directly on one, you'll be met with the venom from its 13 needle-sharp dorsal fin spines. The pain is immediately excruciating, may last for days, and can involve muscular paralysis, breathing difficulties, heart failure, and death.
Pro Tip: Wear sturdy footwear when walking on reef flats or adjacent rocky or weedy areas to avoid this unpleasant experience. It is important to note that stonefish can survive out of the water at low tide. If stung by a stonefish, seek medical attention quickly, as anti-venom is available. There have been no recorded stonefish-related deaths in Australia. One tough Aussie claims he navigated the pain simply by keeping his foot in a bucket of hot water while drinking a bottle of red wine!
Noteworthy, unusual, or dramatic events like shark and crocodile attacks make headlines on global news sites and induce unwarranted fear. However, the animals that create the most risk to our lives are not featured in this list and are, in fact, much more commonplace.
Horses and donkeys cause the most deaths in Australia, mostly related to falls, followed by cattle, generally causing road accidents. Domestic dogs rank as the third biggest killer, predominantly in attacks on children under 4 years of age. Kangaroos causing road accidents and bees stinging allergic victims round out Australia's top five risks to life from animals.
Clearly, these six featured creatures, including sharks, crocodiles, snakes, and spiders, don't warrant the fear surrounding them. With common sense, the chances of encountering these creatures are rare, leaving visitors to enjoy the scenery, weather, and laid-back culture Australia offers. And there's plenty of red wine for medicinal purposes, too!
AustraliaThe Gorgeous Underrated Destination That Needs To Be On Your Australia Travel List

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Plans approved for £161m housing regeneration scheme in London, UK
WCN Editorial Team 20 May 2020 EUROPE BUILDINGS
Brent Council's first virtual planning committee has approved plans for Peel Place, a new £161m housing regeneration scheme at Kilburn in North West London, UK.
Peel Place, the civic quarter of the South Kilburn Masterplan Regeneration Programme, will now proceed with construction.
UK-based developer Countryside Properties and Home Group, a housing association, along with the council, will deliver 308 new tenure-blind homes.
Of the overall housing units, 41% will be for affordable housing, including social rented homes and shared ownership.
Designed by Child Graddon Lewis Architects & Designers, the project will feature a new health centre, a community hub, an affordable gym, an affordable workspace and other retail and commercial spaces.
Countryside, Home Group and Brent Council will offer green spaces and introduce tree-lined avenues and planting throughout the public space.
Situated within walking distance of the Carlton Vale area and a short walk from Queen's Park and Kilburn Park stations, the initial phase of new homes is expected to be completed in the third quarter of 2021.
Countryside Partnerships North & South London managing director Andy Fancy said: "Gaining planning permission for the entire Peel Place scheme is testament to the meticulous work we have done together with Home Group and Brent Council to shape a vision for what's set to become a thriving new community in the capital.
"Having delivered tenure-blind regeneration schemes across the capital for over three decades, the carefully designed homes, community facilities and public amenities will serve the needs and aspirations of local residents, ensuring we have a closely-knit neighbourhood set for the future."
Home Group development executive director Joe Cook said: "We're very pleased indeed to have received approval for what promises to be an excellent regeneration scheme, which will add so much to this North London community.
"Like Countryside, who we work with closely, we have a wealth of experience in delivering successful regeneration schemes in the capital. We are very much looking forward to working with Countryside, Brent Council and the South Kilburn community to deliver homes and amenities that will invigorate and re-energise the community."
The project will deliver 308 new tenure-blind homes. (Credit: Countryside Properties PLC)
Property Alliance, Starwood JV to build £200m mixed-used project in the UK
Tellon Capital to redevelop Cassiobury House at Watford Junction, UK
Winvic selected to deliver two projects at Edgbaston Stadium in UK
NCC to construct Genmab's new headquarters in Denmark
Willmott Dixon wins contract from Cubex to build residential scheme in Bristol, UK
Graham to construct new £80m retirement village in UK
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Mark property or valuables with an identifying mark.
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Celebrate New Year's Eve in Miami aboard our Miami New Year's cruise and enjoy an amazing and unforgettable celebration to bring in 2019. We are going to have you leaving 2018 behind in one of the best ways possible. Celebrate with us and enjoy views of Miami with music blasting all the way into the New Year. Grab your friends and get onboard our Miami New Year's cruise because you definitely don't want to miss out on this cruise experience!
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Enjoy the music, drinks, and food and get ready to end off the year right. Our Miami New Year's cruise will have you bringing in 2019 with a view of the New Year's fireworks as you sip on champagne! Celebrate New Year's Eve with us and enjoy an unforgettable celebration to end 2018 in Miami.


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